| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | yoneda.r | . . 3
⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇) | 
| 2 |  | eqid 2736 | . . . 4
⊢ (𝑄 ×c
𝑂) = (𝑄 ×c 𝑂) | 
| 3 |  | yoneda.q | . . . . 5
⊢ 𝑄 = (𝑂 FuncCat 𝑆) | 
| 4 | 3 | fucbas 18009 | . . . 4
⊢ (𝑂 Func 𝑆) = (Base‘𝑄) | 
| 5 |  | yoneda.o | . . . . 5
⊢ 𝑂 = (oppCat‘𝐶) | 
| 6 |  | yoneda.b | . . . . 5
⊢ 𝐵 = (Base‘𝐶) | 
| 7 | 5, 6 | oppcbas 17762 | . . . 4
⊢ 𝐵 = (Base‘𝑂) | 
| 8 | 2, 4, 7 | xpcbas 18224 | . . 3
⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂)) | 
| 9 |  | eqid 2736 | . . 3
⊢ ((𝑄 ×c
𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇) | 
| 10 |  | yoneda.y | . . . . 5
⊢ 𝑌 = (Yon‘𝐶) | 
| 11 |  | yoneda.1 | . . . . 5
⊢  1 =
(Id‘𝐶) | 
| 12 |  | yoneda.s | . . . . 5
⊢ 𝑆 = (SetCat‘𝑈) | 
| 13 |  | yoneda.t | . . . . 5
⊢ 𝑇 = (SetCat‘𝑉) | 
| 14 |  | yoneda.h | . . . . 5
⊢ 𝐻 =
(HomF‘𝑄) | 
| 15 |  | yoneda.e | . . . . 5
⊢ 𝐸 = (𝑂 evalF 𝑆) | 
| 16 |  | yoneda.z | . . . . 5
⊢ 𝑍 = (𝐻 ∘func
((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉
∘func (𝑄 2ndF 𝑂))
〈,〉F (𝑄 1stF 𝑂))) | 
| 17 |  | yoneda.c | . . . . 5
⊢ (𝜑 → 𝐶 ∈ Cat) | 
| 18 |  | yoneda.w | . . . . 5
⊢ (𝜑 → 𝑉 ∈ 𝑊) | 
| 19 |  | yoneda.u | . . . . 5
⊢ (𝜑 → ran
(Homf ‘𝐶) ⊆ 𝑈) | 
| 20 |  | yoneda.v | . . . . 5
⊢ (𝜑 → (ran
(Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉) | 
| 21 | 10, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 17, 18, 19, 20 | yonedalem1 18318 | . . . 4
⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) | 
| 22 | 21 | simpld 494 | . . 3
⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) | 
| 23 | 21 | simprd 495 | . . 3
⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) | 
| 24 |  | yonedainv.i | . . 3
⊢ 𝐼 = (Inv‘𝑅) | 
| 25 |  | eqid 2736 | . . 3
⊢
(Inv‘𝑇) =
(Inv‘𝑇) | 
| 26 |  | yoneda.m | . . . 4
⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥)))) | 
| 27 | 10, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 17, 18, 19, 20, 26 | yonedalem3 18326 | . . 3
⊢ (𝜑 → 𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸)) | 
| 28 | 17 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → 𝐶 ∈ Cat) | 
| 29 | 18 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → 𝑉 ∈ 𝑊) | 
| 30 | 19 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ran (Homf
‘𝐶) ⊆ 𝑈) | 
| 31 | 20 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ran (Homf
‘𝑄) ∪ 𝑈) ⊆ 𝑉) | 
| 32 |  | simprl 770 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ℎ ∈ (𝑂 Func 𝑆)) | 
| 33 |  | simprr 772 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵) | 
| 34 | 10, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 28, 29, 30, 31, 32, 33, 26 | yonedalem3a 18320 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ((ℎ𝑀𝑤) = (𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤))) ∧ (ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)⟶(ℎ(1st ‘𝐸)𝑤))) | 
| 35 | 34 | simprd 495 | . . . . . . . . 9
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)⟶(ℎ(1st ‘𝐸)𝑤)) | 
| 36 | 28 | adantr 480 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ ((1st ‘ℎ)‘𝑤)) → 𝐶 ∈ Cat) | 
| 37 | 29 | adantr 480 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ ((1st ‘ℎ)‘𝑤)) → 𝑉 ∈ 𝑊) | 
| 38 | 30 | adantr 480 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ ((1st ‘ℎ)‘𝑤)) → ran (Homf
‘𝐶) ⊆ 𝑈) | 
| 39 | 31 | adantr 480 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ ((1st ‘ℎ)‘𝑤)) → (ran (Homf
‘𝑄) ∪ 𝑈) ⊆ 𝑉) | 
| 40 |  | simplrl 776 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ ((1st ‘ℎ)‘𝑤)) → ℎ ∈ (𝑂 Func 𝑆)) | 
| 41 |  | simplrr 777 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ ((1st ‘ℎ)‘𝑤)) → 𝑤 ∈ 𝐵) | 
| 42 |  | yonedainv.n | . . . . . . . . . . . 12
⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))))) | 
| 43 |  | simpr 484 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ ((1st ‘ℎ)‘𝑤)) → 𝑏 ∈ ((1st ‘ℎ)‘𝑤)) | 
| 44 | 10, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 36, 37, 38, 39, 40, 41, 42, 43 | yonedalem4c 18323 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ ((1st ‘ℎ)‘𝑤)) → ((ℎ𝑁𝑤)‘𝑏) ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ)) | 
| 45 | 44 | fmpttd 7134 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (𝑏 ∈ ((1st ‘ℎ)‘𝑤) ↦ ((ℎ𝑁𝑤)‘𝑏)):((1st ‘ℎ)‘𝑤)⟶(((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ)) | 
| 46 | 6 | fvexi 6919 | . . . . . . . . . . . . . . 15
⊢ 𝐵 ∈ V | 
| 47 | 46 | mptex 7244 | . . . . . . . . . . . . . 14
⊢ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢))) ∈ V | 
| 48 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢ (𝑢 ∈ ((1st
‘ℎ)‘𝑤) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢)))) = (𝑢 ∈ ((1st ‘ℎ)‘𝑤) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢)))) | 
| 49 | 47, 48 | fnmpti 6710 | . . . . . . . . . . . . 13
⊢ (𝑢 ∈ ((1st
‘ℎ)‘𝑤) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢)))) Fn ((1st ‘ℎ)‘𝑤) | 
| 50 |  | simpl 482 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 = ℎ ∧ 𝑥 = 𝑤) → 𝑓 = ℎ) | 
| 51 | 50 | fveq2d 6909 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 = ℎ ∧ 𝑥 = 𝑤) → (1st ‘𝑓) = (1st ‘ℎ)) | 
| 52 |  | simpr 484 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 = ℎ ∧ 𝑥 = 𝑤) → 𝑥 = 𝑤) | 
| 53 | 51, 52 | fveq12d 6912 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑓 = ℎ ∧ 𝑥 = 𝑤) → ((1st ‘𝑓)‘𝑥) = ((1st ‘ℎ)‘𝑤)) | 
| 54 |  | simplr 768 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑓 = ℎ ∧ 𝑥 = 𝑤) ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝑤) | 
| 55 | 54 | oveq2d 7448 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓 = ℎ ∧ 𝑥 = 𝑤) ∧ 𝑦 ∈ 𝐵) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐶)𝑤)) | 
| 56 |  | simpll 766 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑓 = ℎ ∧ 𝑥 = 𝑤) ∧ 𝑦 ∈ 𝐵) → 𝑓 = ℎ) | 
| 57 | 56 | fveq2d 6909 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓 = ℎ ∧ 𝑥 = 𝑤) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑓) = (2nd ‘ℎ)) | 
| 58 |  | eqidd 2737 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓 = ℎ ∧ 𝑥 = 𝑤) ∧ 𝑦 ∈ 𝐵) → 𝑦 = 𝑦) | 
| 59 | 57, 54, 58 | oveq123d 7453 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑓 = ℎ ∧ 𝑥 = 𝑤) ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘𝑓)𝑦) = (𝑤(2nd ‘ℎ)𝑦)) | 
| 60 | 59 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑓 = ℎ ∧ 𝑥 = 𝑤) ∧ 𝑦 ∈ 𝐵) → ((𝑥(2nd ‘𝑓)𝑦)‘𝑔) = ((𝑤(2nd ‘ℎ)𝑦)‘𝑔)) | 
| 61 | 60 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓 = ℎ ∧ 𝑥 = 𝑤) ∧ 𝑦 ∈ 𝐵) → (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢) = (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢)) | 
| 62 | 55, 61 | mpteq12dv 5232 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝑓 = ℎ ∧ 𝑥 = 𝑤) ∧ 𝑦 ∈ 𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢))) | 
| 63 | 62 | mpteq2dva 5241 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑓 = ℎ ∧ 𝑥 = 𝑤) → (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢)))) | 
| 64 | 53, 63 | mpteq12dv 5232 | . . . . . . . . . . . . . . . 16
⊢ ((𝑓 = ℎ ∧ 𝑥 = 𝑤) → (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))) = (𝑢 ∈ ((1st ‘ℎ)‘𝑤) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢))))) | 
| 65 |  | fvex 6918 | . . . . . . . . . . . . . . . . 17
⊢
((1st ‘ℎ)‘𝑤) ∈ V | 
| 66 | 65 | mptex 7244 | . . . . . . . . . . . . . . . 16
⊢ (𝑢 ∈ ((1st
‘ℎ)‘𝑤) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢)))) ∈ V | 
| 67 | 64, 42, 66 | ovmpoa 7589 | . . . . . . . . . . . . . . 15
⊢ ((ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵) → (ℎ𝑁𝑤) = (𝑢 ∈ ((1st ‘ℎ)‘𝑤) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢))))) | 
| 68 | 67 | adantl 481 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ𝑁𝑤) = (𝑢 ∈ ((1st ‘ℎ)‘𝑤) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢))))) | 
| 69 | 68 | fneq1d 6660 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ((ℎ𝑁𝑤) Fn ((1st ‘ℎ)‘𝑤) ↔ (𝑢 ∈ ((1st ‘ℎ)‘𝑤) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd ‘ℎ)𝑦)‘𝑔)‘𝑢)))) Fn ((1st ‘ℎ)‘𝑤))) | 
| 70 | 49, 69 | mpbiri 258 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ𝑁𝑤) Fn ((1st ‘ℎ)‘𝑤)) | 
| 71 |  | dffn5 6966 | . . . . . . . . . . . 12
⊢ ((ℎ𝑁𝑤) Fn ((1st ‘ℎ)‘𝑤) ↔ (ℎ𝑁𝑤) = (𝑏 ∈ ((1st ‘ℎ)‘𝑤) ↦ ((ℎ𝑁𝑤)‘𝑏))) | 
| 72 | 70, 71 | sylib 218 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ𝑁𝑤) = (𝑏 ∈ ((1st ‘ℎ)‘𝑤) ↦ ((ℎ𝑁𝑤)‘𝑏))) | 
| 73 | 5 | oppccat 17766 | . . . . . . . . . . . . . 14
⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) | 
| 74 | 17, 73 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑂 ∈ Cat) | 
| 75 | 74 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → 𝑂 ∈ Cat) | 
| 76 | 20 | unssbd 4193 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 ⊆ 𝑉) | 
| 77 | 18, 76 | ssexd 5323 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈 ∈ V) | 
| 78 | 12 | setccat 18131 | . . . . . . . . . . . . . 14
⊢ (𝑈 ∈ V → 𝑆 ∈ Cat) | 
| 79 | 77, 78 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆 ∈ Cat) | 
| 80 | 79 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → 𝑆 ∈ Cat) | 
| 81 | 15, 75, 80, 7, 32, 33 | evlf1 18266 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ(1st ‘𝐸)𝑤) = ((1st ‘ℎ)‘𝑤)) | 
| 82 | 10, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 28, 29, 30, 31, 32, 33 | yonedalem21 18319 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ(1st ‘𝑍)𝑤) = (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ)) | 
| 83 | 72, 81, 82 | feq123d 6724 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ((ℎ𝑁𝑤):(ℎ(1st ‘𝐸)𝑤)⟶(ℎ(1st ‘𝑍)𝑤) ↔ (𝑏 ∈ ((1st ‘ℎ)‘𝑤) ↦ ((ℎ𝑁𝑤)‘𝑏)):((1st ‘ℎ)‘𝑤)⟶(((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ))) | 
| 84 | 45, 83 | mpbird 257 | . . . . . . . . 9
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ𝑁𝑤):(ℎ(1st ‘𝐸)𝑤)⟶(ℎ(1st ‘𝑍)𝑤)) | 
| 85 |  | fcompt 7152 | . . . . . . . . . . 11
⊢ (((ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)⟶(ℎ(1st ‘𝐸)𝑤) ∧ (ℎ𝑁𝑤):(ℎ(1st ‘𝐸)𝑤)⟶(ℎ(1st ‘𝑍)𝑤)) → ((ℎ𝑀𝑤) ∘ (ℎ𝑁𝑤)) = (𝑘 ∈ (ℎ(1st ‘𝐸)𝑤) ↦ ((ℎ𝑀𝑤)‘((ℎ𝑁𝑤)‘𝑘)))) | 
| 86 | 35, 84, 85 | syl2anc 584 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ((ℎ𝑀𝑤) ∘ (ℎ𝑁𝑤)) = (𝑘 ∈ (ℎ(1st ‘𝐸)𝑤) ↦ ((ℎ𝑀𝑤)‘((ℎ𝑁𝑤)‘𝑘)))) | 
| 87 | 81 | eleq2d 2826 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (𝑘 ∈ (ℎ(1st ‘𝐸)𝑤) ↔ 𝑘 ∈ ((1st ‘ℎ)‘𝑤))) | 
| 88 | 87 | biimpa 476 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ (ℎ(1st ‘𝐸)𝑤)) → 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) | 
| 89 | 28 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → 𝐶 ∈ Cat) | 
| 90 | 29 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → 𝑉 ∈ 𝑊) | 
| 91 | 30 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ran (Homf
‘𝐶) ⊆ 𝑈) | 
| 92 | 31 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → (ran (Homf
‘𝑄) ∪ 𝑈) ⊆ 𝑉) | 
| 93 |  | simplrl 776 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ℎ ∈ (𝑂 Func 𝑆)) | 
| 94 |  | simplrr 777 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → 𝑤 ∈ 𝐵) | 
| 95 | 10, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 89, 90, 91, 92, 93, 94, 26 | yonedalem3a 18320 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((ℎ𝑀𝑤) = (𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤))) ∧ (ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)⟶(ℎ(1st ‘𝐸)𝑤))) | 
| 96 | 95 | simpld 494 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → (ℎ𝑀𝑤) = (𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤)))) | 
| 97 | 96 | fveq1d 6907 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((ℎ𝑀𝑤)‘((ℎ𝑁𝑤)‘𝑘)) = ((𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤)))‘((ℎ𝑁𝑤)‘𝑘))) | 
| 98 | 72, 44 | fmpt3d 7135 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ𝑁𝑤):((1st ‘ℎ)‘𝑤)⟶(((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ)) | 
| 99 | 98 | ffvelcdmda 7103 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((ℎ𝑁𝑤)‘𝑘) ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ)) | 
| 100 |  | fveq1 6904 | . . . . . . . . . . . . . . . . 17
⊢ (𝑎 = ((ℎ𝑁𝑤)‘𝑘) → (𝑎‘𝑤) = (((ℎ𝑁𝑤)‘𝑘)‘𝑤)) | 
| 101 | 100 | fveq1d 6907 | . . . . . . . . . . . . . . . 16
⊢ (𝑎 = ((ℎ𝑁𝑤)‘𝑘) → ((𝑎‘𝑤)‘( 1 ‘𝑤)) = ((((ℎ𝑁𝑤)‘𝑘)‘𝑤)‘( 1 ‘𝑤))) | 
| 102 |  | eqid 2736 | . . . . . . . . . . . . . . . 16
⊢ (𝑎 ∈ (((1st
‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤))) = (𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤))) | 
| 103 |  | fvex 6918 | . . . . . . . . . . . . . . . 16
⊢ ((((ℎ𝑁𝑤)‘𝑘)‘𝑤)‘( 1 ‘𝑤)) ∈ V | 
| 104 | 101, 102,
103 | fvmpt 7015 | . . . . . . . . . . . . . . 15
⊢ (((ℎ𝑁𝑤)‘𝑘) ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) → ((𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤)))‘((ℎ𝑁𝑤)‘𝑘)) = ((((ℎ𝑁𝑤)‘𝑘)‘𝑤)‘( 1 ‘𝑤))) | 
| 105 | 99, 104 | syl 17 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤)))‘((ℎ𝑁𝑤)‘𝑘)) = ((((ℎ𝑁𝑤)‘𝑘)‘𝑤)‘( 1 ‘𝑤))) | 
| 106 |  | simpr 484 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) | 
| 107 |  | eqid 2736 | . . . . . . . . . . . . . . . . 17
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) | 
| 108 | 6, 107, 11, 89, 94 | catidcl 17726 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ( 1 ‘𝑤) ∈ (𝑤(Hom ‘𝐶)𝑤)) | 
| 109 | 10, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 89, 90, 91, 92, 93, 94, 42, 106, 94, 108 | yonedalem4b 18322 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((((ℎ𝑁𝑤)‘𝑘)‘𝑤)‘( 1 ‘𝑤)) = (((𝑤(2nd ‘ℎ)𝑤)‘( 1 ‘𝑤))‘𝑘)) | 
| 110 |  | eqid 2736 | . . . . . . . . . . . . . . . . . 18
⊢
(Id‘𝑂) =
(Id‘𝑂) | 
| 111 |  | eqid 2736 | . . . . . . . . . . . . . . . . . 18
⊢
(Id‘𝑆) =
(Id‘𝑆) | 
| 112 |  | relfunc 17908 | . . . . . . . . . . . . . . . . . . 19
⊢ Rel
(𝑂 Func 𝑆) | 
| 113 |  | 1st2ndbr 8068 | . . . . . . . . . . . . . . . . . . 19
⊢ ((Rel
(𝑂 Func 𝑆) ∧ ℎ ∈ (𝑂 Func 𝑆)) → (1st ‘ℎ)(𝑂 Func 𝑆)(2nd ‘ℎ)) | 
| 114 | 112, 93, 113 | sylancr 587 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → (1st ‘ℎ)(𝑂 Func 𝑆)(2nd ‘ℎ)) | 
| 115 | 7, 110, 111, 114, 94 | funcid 17916 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((𝑤(2nd ‘ℎ)𝑤)‘((Id‘𝑂)‘𝑤)) = ((Id‘𝑆)‘((1st ‘ℎ)‘𝑤))) | 
| 116 | 5, 11 | oppcid 17765 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝐶 ∈ Cat →
(Id‘𝑂) = 1
) | 
| 117 | 89, 116 | syl 17 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → (Id‘𝑂) = 1 ) | 
| 118 | 117 | fveq1d 6907 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((Id‘𝑂)‘𝑤) = ( 1 ‘𝑤)) | 
| 119 | 118 | fveq2d 6909 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((𝑤(2nd ‘ℎ)𝑤)‘((Id‘𝑂)‘𝑤)) = ((𝑤(2nd ‘ℎ)𝑤)‘( 1 ‘𝑤))) | 
| 120 | 77 | ad2antrr 726 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → 𝑈 ∈ V) | 
| 121 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(Base‘𝑆) =
(Base‘𝑆) | 
| 122 | 7, 121, 114 | funcf1 17912 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → (1st ‘ℎ):𝐵⟶(Base‘𝑆)) | 
| 123 | 12, 120 | setcbas 18124 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → 𝑈 = (Base‘𝑆)) | 
| 124 | 123 | feq3d 6722 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((1st ‘ℎ):𝐵⟶𝑈 ↔ (1st ‘ℎ):𝐵⟶(Base‘𝑆))) | 
| 125 | 122, 124 | mpbird 257 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → (1st ‘ℎ):𝐵⟶𝑈) | 
| 126 | 125, 94 | ffvelcdmd 7104 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((1st ‘ℎ)‘𝑤) ∈ 𝑈) | 
| 127 | 12, 111, 120, 126 | setcid 18132 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((Id‘𝑆)‘((1st ‘ℎ)‘𝑤)) = ( I ↾ ((1st
‘ℎ)‘𝑤))) | 
| 128 | 115, 119,
127 | 3eqtr3d 2784 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((𝑤(2nd ‘ℎ)𝑤)‘( 1 ‘𝑤)) = ( I ↾ ((1st
‘ℎ)‘𝑤))) | 
| 129 | 128 | fveq1d 6907 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → (((𝑤(2nd ‘ℎ)𝑤)‘( 1 ‘𝑤))‘𝑘) = (( I ↾ ((1st
‘ℎ)‘𝑤))‘𝑘)) | 
| 130 |  | fvresi 7194 | . . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ((1st
‘ℎ)‘𝑤) → (( I ↾
((1st ‘ℎ)‘𝑤))‘𝑘) = 𝑘) | 
| 131 | 130 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → (( I ↾ ((1st
‘ℎ)‘𝑤))‘𝑘) = 𝑘) | 
| 132 | 109, 129,
131 | 3eqtrd 2780 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((((ℎ𝑁𝑤)‘𝑘)‘𝑤)‘( 1 ‘𝑤)) = 𝑘) | 
| 133 | 97, 105, 132 | 3eqtrd 2780 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ ((1st ‘ℎ)‘𝑤)) → ((ℎ𝑀𝑤)‘((ℎ𝑁𝑤)‘𝑘)) = 𝑘) | 
| 134 | 88, 133 | syldan 591 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑘 ∈ (ℎ(1st ‘𝐸)𝑤)) → ((ℎ𝑀𝑤)‘((ℎ𝑁𝑤)‘𝑘)) = 𝑘) | 
| 135 | 134 | mpteq2dva 5241 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (𝑘 ∈ (ℎ(1st ‘𝐸)𝑤) ↦ ((ℎ𝑀𝑤)‘((ℎ𝑁𝑤)‘𝑘))) = (𝑘 ∈ (ℎ(1st ‘𝐸)𝑤) ↦ 𝑘)) | 
| 136 |  | mptresid 6068 | . . . . . . . . . . 11
⊢ ( I
↾ (ℎ(1st
‘𝐸)𝑤)) = (𝑘 ∈ (ℎ(1st ‘𝐸)𝑤) ↦ 𝑘) | 
| 137 | 135, 136 | eqtr4di 2794 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (𝑘 ∈ (ℎ(1st ‘𝐸)𝑤) ↦ ((ℎ𝑀𝑤)‘((ℎ𝑁𝑤)‘𝑘))) = ( I ↾ (ℎ(1st ‘𝐸)𝑤))) | 
| 138 | 86, 137 | eqtrd 2776 | . . . . . . . . 9
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ((ℎ𝑀𝑤) ∘ (ℎ𝑁𝑤)) = ( I ↾ (ℎ(1st ‘𝐸)𝑤))) | 
| 139 |  | fcompt 7152 | . . . . . . . . . . 11
⊢ (((ℎ𝑁𝑤):(ℎ(1st ‘𝐸)𝑤)⟶(ℎ(1st ‘𝑍)𝑤) ∧ (ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)⟶(ℎ(1st ‘𝐸)𝑤)) → ((ℎ𝑁𝑤) ∘ (ℎ𝑀𝑤)) = (𝑏 ∈ (ℎ(1st ‘𝑍)𝑤) ↦ ((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏)))) | 
| 140 | 84, 35, 139 | syl2anc 584 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ((ℎ𝑁𝑤) ∘ (ℎ𝑀𝑤)) = (𝑏 ∈ (ℎ(1st ‘𝑍)𝑤) ↦ ((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏)))) | 
| 141 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆) | 
| 142 | 28 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → 𝐶 ∈ Cat) | 
| 143 | 29 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → 𝑉 ∈ 𝑊) | 
| 144 | 30 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ran (Homf
‘𝐶) ⊆ 𝑈) | 
| 145 | 31 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → (ran (Homf
‘𝑄) ∪ 𝑈) ⊆ 𝑉) | 
| 146 |  | simplrl 776 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ℎ ∈ (𝑂 Func 𝑆)) | 
| 147 |  | simplrr 777 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → 𝑤 ∈ 𝐵) | 
| 148 | 81 | feq3d 6722 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ((ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)⟶(ℎ(1st ‘𝐸)𝑤) ↔ (ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)⟶((1st ‘ℎ)‘𝑤))) | 
| 149 | 35, 148 | mpbid 232 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)⟶((1st ‘ℎ)‘𝑤)) | 
| 150 | 149 | ffvelcdmda 7103 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((ℎ𝑀𝑤)‘𝑏) ∈ ((1st ‘ℎ)‘𝑤)) | 
| 151 | 10, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 142, 143, 144, 145, 146, 147, 42, 150 | yonedalem4c 18323 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏)) ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ)) | 
| 152 | 141, 151 | nat1st2nd 18000 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏)) ∈ (〈(1st
‘((1st ‘𝑌)‘𝑤)), (2nd ‘((1st
‘𝑌)‘𝑤))〉(𝑂 Nat 𝑆)〈(1st ‘ℎ), (2nd ‘ℎ)〉)) | 
| 153 | 141, 152,
7 | natfn 18003 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏)) Fn 𝐵) | 
| 154 | 82 | eleq2d 2826 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (𝑏 ∈ (ℎ(1st ‘𝑍)𝑤) ↔ 𝑏 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ))) | 
| 155 | 154 | biimpa 476 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → 𝑏 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ)) | 
| 156 | 141, 155 | nat1st2nd 18000 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → 𝑏 ∈ (〈(1st
‘((1st ‘𝑌)‘𝑤)), (2nd ‘((1st
‘𝑌)‘𝑤))〉(𝑂 Nat 𝑆)〈(1st ‘ℎ), (2nd ‘ℎ)〉)) | 
| 157 | 141, 156,
7 | natfn 18003 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → 𝑏 Fn 𝐵) | 
| 158 | 142 | adantr 480 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → 𝐶 ∈ Cat) | 
| 159 | 147 | adantr 480 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → 𝑤 ∈ 𝐵) | 
| 160 |  | simpr 484 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵) | 
| 161 | 10, 6, 158, 159, 107, 160 | yon11 18310 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤)) | 
| 162 | 161 | eleq2d 2826 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ↔ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤))) | 
| 163 | 162 | biimpa 476 | . . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) | 
| 164 | 158 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat) | 
| 165 | 143 | ad2antrr 726 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉 ∈ 𝑊) | 
| 166 | 144 | ad2antrr 726 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Homf
‘𝐶) ⊆ 𝑈) | 
| 167 | 145 | ad2antrr 726 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Homf
‘𝑄) ∪ 𝑈) ⊆ 𝑉) | 
| 168 | 146 | ad2antrr 726 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ℎ ∈ (𝑂 Func 𝑆)) | 
| 169 | 159 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑤 ∈ 𝐵) | 
| 170 | 150 | ad2antrr 726 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((ℎ𝑀𝑤)‘𝑏) ∈ ((1st ‘ℎ)‘𝑤)) | 
| 171 |  | simplr 768 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧 ∈ 𝐵) | 
| 172 |  | simpr 484 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) | 
| 173 | 10, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 164, 165, 166, 167, 168, 169, 42, 170, 171, 172 | yonedalem4b 18322 | . . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))‘𝑧)‘𝑘) = (((𝑤(2nd ‘ℎ)𝑧)‘𝑘)‘((ℎ𝑀𝑤)‘𝑏))) | 
| 174 | 10, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 164, 165, 166, 167, 168, 169, 26 | yonedalem3a 18320 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((ℎ𝑀𝑤) = (𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤))) ∧ (ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)⟶(ℎ(1st ‘𝐸)𝑤))) | 
| 175 | 174 | simpld 494 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ℎ𝑀𝑤) = (𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤)))) | 
| 176 | 175 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((ℎ𝑀𝑤)‘𝑏) = ((𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤)))‘𝑏)) | 
| 177 | 155 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑏 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ)) | 
| 178 |  | fveq1 6904 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 = 𝑏 → (𝑎‘𝑤) = (𝑏‘𝑤)) | 
| 179 | 178 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎 = 𝑏 → ((𝑎‘𝑤)‘( 1 ‘𝑤)) = ((𝑏‘𝑤)‘( 1 ‘𝑤))) | 
| 180 |  | fvex 6918 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑏‘𝑤)‘( 1 ‘𝑤)) ∈ V | 
| 181 | 179, 102,
180 | fvmpt 7015 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑏 ∈ (((1st
‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) → ((𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤)))‘𝑏) = ((𝑏‘𝑤)‘( 1 ‘𝑤))) | 
| 182 | 177, 181 | syl 17 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑎 ∈ (((1st ‘𝑌)‘𝑤)(𝑂 Nat 𝑆)ℎ) ↦ ((𝑎‘𝑤)‘( 1 ‘𝑤)))‘𝑏) = ((𝑏‘𝑤)‘( 1 ‘𝑤))) | 
| 183 | 176, 182 | eqtrd 2776 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((ℎ𝑀𝑤)‘𝑏) = ((𝑏‘𝑤)‘( 1 ‘𝑤))) | 
| 184 | 183 | fveq2d 6909 | . . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘ℎ)𝑧)‘𝑘)‘((ℎ𝑀𝑤)‘𝑏)) = (((𝑤(2nd ‘ℎ)𝑧)‘𝑘)‘((𝑏‘𝑤)‘( 1 ‘𝑤)))) | 
| 185 | 156 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑏 ∈ (〈(1st
‘((1st ‘𝑌)‘𝑤)), (2nd ‘((1st
‘𝑌)‘𝑤))〉(𝑂 Nat 𝑆)〈(1st ‘ℎ), (2nd ‘ℎ)〉)) | 
| 186 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (Hom
‘𝑂) = (Hom
‘𝑂) | 
| 187 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(comp‘𝑆) =
(comp‘𝑆) | 
| 188 | 107, 5 | oppchom 17759 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑤(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑤) | 
| 189 | 172, 188 | eleqtrrdi 2851 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑘 ∈ (𝑤(Hom ‘𝑂)𝑧)) | 
| 190 | 141, 185,
7, 186, 187, 169, 171, 189 | nati 18004 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏‘𝑧)(〈((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤), ((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧)〉(comp‘𝑆)((1st ‘ℎ)‘𝑧))((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘)) = (((𝑤(2nd ‘ℎ)𝑧)‘𝑘)(〈((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤), ((1st ‘ℎ)‘𝑤)〉(comp‘𝑆)((1st ‘ℎ)‘𝑧))(𝑏‘𝑤))) | 
| 191 | 77 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → 𝑈 ∈ V) | 
| 192 | 191 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → 𝑈 ∈ V) | 
| 193 | 192 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑈 ∈ V) | 
| 194 |  | relfunc 17908 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ Rel
(𝐶 Func 𝑄) | 
| 195 | 10, 17, 5, 12, 3, 77, 19 | yoncl 18308 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄)) | 
| 196 |  | 1st2ndbr 8068 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((Rel
(𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) | 
| 197 | 194, 195,
196 | sylancr 587 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → (1st
‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) | 
| 198 | 6, 4, 197 | funcf1 17912 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → (1st
‘𝑌):𝐵⟶(𝑂 Func 𝑆)) | 
| 199 | 198 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆)) | 
| 200 | 199, 147 | ffvelcdmd 7104 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) | 
| 201 |  | 1st2ndbr 8068 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((Rel
(𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (1st
‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑤))) | 
| 202 | 112, 200,
201 | sylancr 587 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → (1st
‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑤))) | 
| 203 | 7, 121, 202 | funcf1 17912 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → (1st
‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆)) | 
| 204 | 12, 191 | setcbas 18124 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → 𝑈 = (Base‘𝑆)) | 
| 205 | 204 | feq3d 6722 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((1st
‘((1st ‘𝑌)‘𝑤)):𝐵⟶𝑈 ↔ (1st
‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))) | 
| 206 | 203, 205 | mpbird 257 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → (1st
‘((1st ‘𝑌)‘𝑤)):𝐵⟶𝑈) | 
| 207 | 206, 147 | ffvelcdmd 7104 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤) ∈ 𝑈) | 
| 208 | 207 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤) ∈ 𝑈) | 
| 209 | 206 | ffvelcdmda 7103 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ 𝑈) | 
| 210 | 209 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ 𝑈) | 
| 211 | 112, 146,
113 | sylancr 587 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → (1st ‘ℎ)(𝑂 Func 𝑆)(2nd ‘ℎ)) | 
| 212 | 7, 121, 211 | funcf1 17912 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → (1st ‘ℎ):𝐵⟶(Base‘𝑆)) | 
| 213 | 204 | feq3d 6722 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((1st ‘ℎ):𝐵⟶𝑈 ↔ (1st ‘ℎ):𝐵⟶(Base‘𝑆))) | 
| 214 | 212, 213 | mpbird 257 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → (1st ‘ℎ):𝐵⟶𝑈) | 
| 215 | 214 | ffvelcdmda 7103 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → ((1st ‘ℎ)‘𝑧) ∈ 𝑈) | 
| 216 | 215 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘ℎ)‘𝑧) ∈ 𝑈) | 
| 217 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (Hom
‘𝑆) = (Hom
‘𝑆) | 
| 218 | 202 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st
‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑤))) | 
| 219 | 7, 186, 217, 218, 169, 171 | funcf2 17914 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧):(𝑤(Hom ‘𝑂)𝑧)⟶(((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧))) | 
| 220 | 219, 189 | ffvelcdmd 7104 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘) ∈ (((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧))) | 
| 221 | 12, 193, 217, 208, 210 | elsetchom 18127 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘) ∈ (((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧)) ↔ ((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘):((1st ‘((1st
‘𝑌)‘𝑤))‘𝑤)⟶((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧))) | 
| 222 | 220, 221 | mpbid 232 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘):((1st ‘((1st
‘𝑌)‘𝑤))‘𝑤)⟶((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧)) | 
| 223 | 156 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → 𝑏 ∈ (〈(1st
‘((1st ‘𝑌)‘𝑤)), (2nd ‘((1st
‘𝑌)‘𝑤))〉(𝑂 Nat 𝑆)〈(1st ‘ℎ), (2nd ‘ℎ)〉)) | 
| 224 | 141, 223,
7, 217, 160 | natcl 18002 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → (𝑏‘𝑧) ∈ (((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st ‘ℎ)‘𝑧))) | 
| 225 | 12, 192, 217, 209, 215 | elsetchom 18127 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → ((𝑏‘𝑧) ∈ (((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st ‘ℎ)‘𝑧)) ↔ (𝑏‘𝑧):((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧)⟶((1st ‘ℎ)‘𝑧))) | 
| 226 | 224, 225 | mpbid 232 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → (𝑏‘𝑧):((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧)⟶((1st ‘ℎ)‘𝑧)) | 
| 227 | 226 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑏‘𝑧):((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧)⟶((1st ‘ℎ)‘𝑧)) | 
| 228 | 12, 193, 187, 208, 210, 216, 222, 227 | setcco 18129 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏‘𝑧)(〈((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤), ((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧)〉(comp‘𝑆)((1st ‘ℎ)‘𝑧))((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘)) = ((𝑏‘𝑧) ∘ ((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘))) | 
| 229 | 214, 147 | ffvelcdmd 7104 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((1st ‘ℎ)‘𝑤) ∈ 𝑈) | 
| 230 | 229 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘ℎ)‘𝑤) ∈ 𝑈) | 
| 231 | 141, 156,
7, 217, 147 | natcl 18002 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → (𝑏‘𝑤) ∈ (((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st ‘ℎ)‘𝑤))) | 
| 232 | 12, 191, 217, 207, 229 | elsetchom 18127 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((𝑏‘𝑤) ∈ (((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st ‘ℎ)‘𝑤)) ↔ (𝑏‘𝑤):((1st ‘((1st
‘𝑌)‘𝑤))‘𝑤)⟶((1st ‘ℎ)‘𝑤))) | 
| 233 | 231, 232 | mpbid 232 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → (𝑏‘𝑤):((1st ‘((1st
‘𝑌)‘𝑤))‘𝑤)⟶((1st ‘ℎ)‘𝑤)) | 
| 234 | 233 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑏‘𝑤):((1st ‘((1st
‘𝑌)‘𝑤))‘𝑤)⟶((1st ‘ℎ)‘𝑤)) | 
| 235 | 112, 168,
113 | sylancr 587 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘ℎ)(𝑂 Func 𝑆)(2nd ‘ℎ)) | 
| 236 | 7, 186, 217, 235, 169, 171 | funcf2 17914 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑤(2nd ‘ℎ)𝑧):(𝑤(Hom ‘𝑂)𝑧)⟶(((1st ‘ℎ)‘𝑤)(Hom ‘𝑆)((1st ‘ℎ)‘𝑧))) | 
| 237 | 236, 189 | ffvelcdmd 7104 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd ‘ℎ)𝑧)‘𝑘) ∈ (((1st ‘ℎ)‘𝑤)(Hom ‘𝑆)((1st ‘ℎ)‘𝑧))) | 
| 238 | 12, 193, 217, 230, 216 | elsetchom 18127 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘ℎ)𝑧)‘𝑘) ∈ (((1st ‘ℎ)‘𝑤)(Hom ‘𝑆)((1st ‘ℎ)‘𝑧)) ↔ ((𝑤(2nd ‘ℎ)𝑧)‘𝑘):((1st ‘ℎ)‘𝑤)⟶((1st ‘ℎ)‘𝑧))) | 
| 239 | 237, 238 | mpbid 232 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd ‘ℎ)𝑧)‘𝑘):((1st ‘ℎ)‘𝑤)⟶((1st ‘ℎ)‘𝑧)) | 
| 240 | 12, 193, 187, 208, 230, 216, 234, 239 | setcco 18129 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘ℎ)𝑧)‘𝑘)(〈((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤), ((1st ‘ℎ)‘𝑤)〉(comp‘𝑆)((1st ‘ℎ)‘𝑧))(𝑏‘𝑤)) = (((𝑤(2nd ‘ℎ)𝑧)‘𝑘) ∘ (𝑏‘𝑤))) | 
| 241 | 190, 228,
240 | 3eqtr3d 2784 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏‘𝑧) ∘ ((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘)) = (((𝑤(2nd ‘ℎ)𝑧)‘𝑘) ∘ (𝑏‘𝑤))) | 
| 242 | 241 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑏‘𝑧) ∘ ((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘))‘( 1 ‘𝑤)) = ((((𝑤(2nd ‘ℎ)𝑧)‘𝑘) ∘ (𝑏‘𝑤))‘( 1 ‘𝑤))) | 
| 243 | 6, 107, 11, 142, 147 | catidcl 17726 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ( 1 ‘𝑤) ∈ (𝑤(Hom ‘𝐶)𝑤)) | 
| 244 | 10, 6, 142, 147, 107, 147 | yon11 18310 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤) = (𝑤(Hom ‘𝐶)𝑤)) | 
| 245 | 243, 244 | eleqtrrd 2843 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ( 1 ‘𝑤) ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤)) | 
| 246 | 245 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ( 1 ‘𝑤) ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑤)) | 
| 247 | 222, 246 | fvco3d 7008 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑏‘𝑧) ∘ ((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘))‘( 1 ‘𝑤)) = ((𝑏‘𝑧)‘(((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘)‘( 1 ‘𝑤)))) | 
| 248 | 233, 245 | fvco3d 7008 | . . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((((𝑤(2nd ‘ℎ)𝑧)‘𝑘) ∘ (𝑏‘𝑤))‘( 1 ‘𝑤)) = (((𝑤(2nd ‘ℎ)𝑧)‘𝑘)‘((𝑏‘𝑤)‘( 1 ‘𝑤)))) | 
| 249 | 248 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((𝑤(2nd ‘ℎ)𝑧)‘𝑘) ∘ (𝑏‘𝑤))‘( 1 ‘𝑤)) = (((𝑤(2nd ‘ℎ)𝑧)‘𝑘)‘((𝑏‘𝑤)‘( 1 ‘𝑤)))) | 
| 250 | 242, 247,
249 | 3eqtr3d 2784 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏‘𝑧)‘(((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘)‘( 1 ‘𝑤))) = (((𝑤(2nd ‘ℎ)𝑧)‘𝑘)‘((𝑏‘𝑤)‘( 1 ‘𝑤)))) | 
| 251 |  | eqid 2736 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(comp‘𝐶) =
(comp‘𝐶) | 
| 252 | 243 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ( 1 ‘𝑤) ∈ (𝑤(Hom ‘𝐶)𝑤)) | 
| 253 | 10, 6, 164, 169, 107, 169, 251, 171, 172, 252 | yon12 18311 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘)‘( 1 ‘𝑤)) = (( 1 ‘𝑤)(〈𝑧, 𝑤〉(comp‘𝐶)𝑤)𝑘)) | 
| 254 | 6, 107, 11, 164, 171, 251, 169, 172 | catlid 17727 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (( 1 ‘𝑤)(〈𝑧, 𝑤〉(comp‘𝐶)𝑤)𝑘) = 𝑘) | 
| 255 | 253, 254 | eqtrd 2776 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘)‘( 1 ‘𝑤)) = 𝑘) | 
| 256 | 255 | fveq2d 6909 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏‘𝑧)‘(((𝑤(2nd ‘((1st
‘𝑌)‘𝑤))𝑧)‘𝑘)‘( 1 ‘𝑤))) = ((𝑏‘𝑧)‘𝑘)) | 
| 257 | 250, 256 | eqtr3d 2778 | . . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘ℎ)𝑧)‘𝑘)‘((𝑏‘𝑤)‘( 1 ‘𝑤))) = ((𝑏‘𝑧)‘𝑘)) | 
| 258 | 173, 184,
257 | 3eqtrd 2780 | . . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))‘𝑧)‘𝑘) = ((𝑏‘𝑧)‘𝑘)) | 
| 259 | 163, 258 | syldan 591 | . . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧)) → ((((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))‘𝑧)‘𝑘) = ((𝑏‘𝑧)‘𝑘)) | 
| 260 | 259 | mpteq2dva 5241 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ↦ ((((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))‘𝑧)‘𝑘)) = (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ↦ ((𝑏‘𝑧)‘𝑘))) | 
| 261 | 152 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → ((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏)) ∈ (〈(1st
‘((1st ‘𝑌)‘𝑤)), (2nd ‘((1st
‘𝑌)‘𝑤))〉(𝑂 Nat 𝑆)〈(1st ‘ℎ), (2nd ‘ℎ)〉)) | 
| 262 | 141, 261,
7, 217, 160 | natcl 18002 | . . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → (((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))‘𝑧) ∈ (((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st ‘ℎ)‘𝑧))) | 
| 263 | 12, 192, 217, 209, 215 | elsetchom 18127 | . . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → ((((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))‘𝑧) ∈ (((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st ‘ℎ)‘𝑧)) ↔ (((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))‘𝑧):((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧)⟶((1st ‘ℎ)‘𝑧))) | 
| 264 | 262, 263 | mpbid 232 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → (((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))‘𝑧):((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧)⟶((1st ‘ℎ)‘𝑧)) | 
| 265 | 264 | feqmptd 6976 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → (((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))‘𝑧) = (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ↦ ((((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))‘𝑧)‘𝑘))) | 
| 266 | 226 | feqmptd 6976 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → (𝑏‘𝑧) = (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ↦ ((𝑏‘𝑧)‘𝑘))) | 
| 267 | 260, 265,
266 | 3eqtr4d 2786 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) ∧ 𝑧 ∈ 𝐵) → (((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))‘𝑧) = (𝑏‘𝑧)) | 
| 268 | 153, 157,
267 | eqfnfvd 7053 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) ∧ 𝑏 ∈ (ℎ(1st ‘𝑍)𝑤)) → ((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏)) = 𝑏) | 
| 269 | 268 | mpteq2dva 5241 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (𝑏 ∈ (ℎ(1st ‘𝑍)𝑤) ↦ ((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))) = (𝑏 ∈ (ℎ(1st ‘𝑍)𝑤) ↦ 𝑏)) | 
| 270 |  | mptresid 6068 | . . . . . . . . . . 11
⊢ ( I
↾ (ℎ(1st
‘𝑍)𝑤)) = (𝑏 ∈ (ℎ(1st ‘𝑍)𝑤) ↦ 𝑏) | 
| 271 | 269, 270 | eqtr4di 2794 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (𝑏 ∈ (ℎ(1st ‘𝑍)𝑤) ↦ ((ℎ𝑁𝑤)‘((ℎ𝑀𝑤)‘𝑏))) = ( I ↾ (ℎ(1st ‘𝑍)𝑤))) | 
| 272 | 140, 271 | eqtrd 2776 | . . . . . . . . 9
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ((ℎ𝑁𝑤) ∘ (ℎ𝑀𝑤)) = ( I ↾ (ℎ(1st ‘𝑍)𝑤))) | 
| 273 |  | fcof1o 7317 | . . . . . . . . 9
⊢ ((((ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)⟶(ℎ(1st ‘𝐸)𝑤) ∧ (ℎ𝑁𝑤):(ℎ(1st ‘𝐸)𝑤)⟶(ℎ(1st ‘𝑍)𝑤)) ∧ (((ℎ𝑀𝑤) ∘ (ℎ𝑁𝑤)) = ( I ↾ (ℎ(1st ‘𝐸)𝑤)) ∧ ((ℎ𝑁𝑤) ∘ (ℎ𝑀𝑤)) = ( I ↾ (ℎ(1st ‘𝑍)𝑤)))) → ((ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)–1-1-onto→(ℎ(1st ‘𝐸)𝑤) ∧ ◡(ℎ𝑀𝑤) = (ℎ𝑁𝑤))) | 
| 274 | 35, 84, 138, 272, 273 | syl22anc 838 | . . . . . . . 8
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ((ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)–1-1-onto→(ℎ(1st ‘𝐸)𝑤) ∧ ◡(ℎ𝑀𝑤) = (ℎ𝑁𝑤))) | 
| 275 |  | eqcom 2743 | . . . . . . . . 9
⊢ (◡(ℎ𝑀𝑤) = (ℎ𝑁𝑤) ↔ (ℎ𝑁𝑤) = ◡(ℎ𝑀𝑤)) | 
| 276 | 275 | anbi2i 623 | . . . . . . . 8
⊢ (((ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)–1-1-onto→(ℎ(1st ‘𝐸)𝑤) ∧ ◡(ℎ𝑀𝑤) = (ℎ𝑁𝑤)) ↔ ((ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)–1-1-onto→(ℎ(1st ‘𝐸)𝑤) ∧ (ℎ𝑁𝑤) = ◡(ℎ𝑀𝑤))) | 
| 277 | 274, 276 | sylib 218 | . . . . . . 7
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ((ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)–1-1-onto→(ℎ(1st ‘𝐸)𝑤) ∧ (ℎ𝑁𝑤) = ◡(ℎ𝑀𝑤))) | 
| 278 |  | eqid 2736 | . . . . . . . . . . 11
⊢
(Base‘𝑇) =
(Base‘𝑇) | 
| 279 |  | relfunc 17908 | . . . . . . . . . . . 12
⊢ Rel
((𝑄
×c 𝑂) Func 𝑇) | 
| 280 |  | 1st2ndbr 8068 | . . . . . . . . . . . 12
⊢ ((Rel
((𝑄
×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st ‘𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd ‘𝑍)) | 
| 281 | 279, 22, 280 | sylancr 587 | . . . . . . . . . . 11
⊢ (𝜑 → (1st
‘𝑍)((𝑄 ×c
𝑂) Func 𝑇)(2nd ‘𝑍)) | 
| 282 | 8, 278, 281 | funcf1 17912 | . . . . . . . . . 10
⊢ (𝜑 → (1st
‘𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇)) | 
| 283 | 13, 18 | setcbas 18124 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑉 = (Base‘𝑇)) | 
| 284 | 283 | feq3d 6722 | . . . . . . . . . 10
⊢ (𝜑 → ((1st
‘𝑍):((𝑂 Func 𝑆) × 𝐵)⟶𝑉 ↔ (1st ‘𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))) | 
| 285 | 282, 284 | mpbird 257 | . . . . . . . . 9
⊢ (𝜑 → (1st
‘𝑍):((𝑂 Func 𝑆) × 𝐵)⟶𝑉) | 
| 286 | 285 | fovcdmda 7605 | . . . . . . . 8
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ(1st ‘𝑍)𝑤) ∈ 𝑉) | 
| 287 |  | 1st2ndbr 8068 | . . . . . . . . . . . 12
⊢ ((Rel
((𝑄
×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st ‘𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd ‘𝐸)) | 
| 288 | 279, 23, 287 | sylancr 587 | . . . . . . . . . . 11
⊢ (𝜑 → (1st
‘𝐸)((𝑄 ×c
𝑂) Func 𝑇)(2nd ‘𝐸)) | 
| 289 | 8, 278, 288 | funcf1 17912 | . . . . . . . . . 10
⊢ (𝜑 → (1st
‘𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇)) | 
| 290 | 283 | feq3d 6722 | . . . . . . . . . 10
⊢ (𝜑 → ((1st
‘𝐸):((𝑂 Func 𝑆) × 𝐵)⟶𝑉 ↔ (1st ‘𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))) | 
| 291 | 289, 290 | mpbird 257 | . . . . . . . . 9
⊢ (𝜑 → (1st
‘𝐸):((𝑂 Func 𝑆) × 𝐵)⟶𝑉) | 
| 292 | 291 | fovcdmda 7605 | . . . . . . . 8
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ(1st ‘𝐸)𝑤) ∈ 𝑉) | 
| 293 | 13, 29, 286, 292, 25 | setcinv 18136 | . . . . . . 7
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → ((ℎ𝑀𝑤)((ℎ(1st ‘𝑍)𝑤)(Inv‘𝑇)(ℎ(1st ‘𝐸)𝑤))(ℎ𝑁𝑤) ↔ ((ℎ𝑀𝑤):(ℎ(1st ‘𝑍)𝑤)–1-1-onto→(ℎ(1st ‘𝐸)𝑤) ∧ (ℎ𝑁𝑤) = ◡(ℎ𝑀𝑤)))) | 
| 294 | 277, 293 | mpbird 257 | . . . . . 6
⊢ ((𝜑 ∧ (ℎ ∈ (𝑂 Func 𝑆) ∧ 𝑤 ∈ 𝐵)) → (ℎ𝑀𝑤)((ℎ(1st ‘𝑍)𝑤)(Inv‘𝑇)(ℎ(1st ‘𝐸)𝑤))(ℎ𝑁𝑤)) | 
| 295 | 294 | ralrimivva 3201 | . . . . 5
⊢ (𝜑 → ∀ℎ ∈ (𝑂 Func 𝑆)∀𝑤 ∈ 𝐵 (ℎ𝑀𝑤)((ℎ(1st ‘𝑍)𝑤)(Inv‘𝑇)(ℎ(1st ‘𝐸)𝑤))(ℎ𝑁𝑤)) | 
| 296 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑧 = 〈ℎ, 𝑤〉 → (𝑀‘𝑧) = (𝑀‘〈ℎ, 𝑤〉)) | 
| 297 |  | df-ov 7435 | . . . . . . . 8
⊢ (ℎ𝑀𝑤) = (𝑀‘〈ℎ, 𝑤〉) | 
| 298 | 296, 297 | eqtr4di 2794 | . . . . . . 7
⊢ (𝑧 = 〈ℎ, 𝑤〉 → (𝑀‘𝑧) = (ℎ𝑀𝑤)) | 
| 299 |  | fveq2 6905 | . . . . . . . . 9
⊢ (𝑧 = 〈ℎ, 𝑤〉 → ((1st ‘𝑍)‘𝑧) = ((1st ‘𝑍)‘〈ℎ, 𝑤〉)) | 
| 300 |  | df-ov 7435 | . . . . . . . . 9
⊢ (ℎ(1st ‘𝑍)𝑤) = ((1st ‘𝑍)‘〈ℎ, 𝑤〉) | 
| 301 | 299, 300 | eqtr4di 2794 | . . . . . . . 8
⊢ (𝑧 = 〈ℎ, 𝑤〉 → ((1st ‘𝑍)‘𝑧) = (ℎ(1st ‘𝑍)𝑤)) | 
| 302 |  | fveq2 6905 | . . . . . . . . 9
⊢ (𝑧 = 〈ℎ, 𝑤〉 → ((1st ‘𝐸)‘𝑧) = ((1st ‘𝐸)‘〈ℎ, 𝑤〉)) | 
| 303 |  | df-ov 7435 | . . . . . . . . 9
⊢ (ℎ(1st ‘𝐸)𝑤) = ((1st ‘𝐸)‘〈ℎ, 𝑤〉) | 
| 304 | 302, 303 | eqtr4di 2794 | . . . . . . . 8
⊢ (𝑧 = 〈ℎ, 𝑤〉 → ((1st ‘𝐸)‘𝑧) = (ℎ(1st ‘𝐸)𝑤)) | 
| 305 | 301, 304 | oveq12d 7450 | . . . . . . 7
⊢ (𝑧 = 〈ℎ, 𝑤〉 → (((1st ‘𝑍)‘𝑧)(Inv‘𝑇)((1st ‘𝐸)‘𝑧)) = ((ℎ(1st ‘𝑍)𝑤)(Inv‘𝑇)(ℎ(1st ‘𝐸)𝑤))) | 
| 306 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑧 = 〈ℎ, 𝑤〉 → (𝑁‘𝑧) = (𝑁‘〈ℎ, 𝑤〉)) | 
| 307 |  | df-ov 7435 | . . . . . . . 8
⊢ (ℎ𝑁𝑤) = (𝑁‘〈ℎ, 𝑤〉) | 
| 308 | 306, 307 | eqtr4di 2794 | . . . . . . 7
⊢ (𝑧 = 〈ℎ, 𝑤〉 → (𝑁‘𝑧) = (ℎ𝑁𝑤)) | 
| 309 | 298, 305,
308 | breq123d 5156 | . . . . . 6
⊢ (𝑧 = 〈ℎ, 𝑤〉 → ((𝑀‘𝑧)(((1st ‘𝑍)‘𝑧)(Inv‘𝑇)((1st ‘𝐸)‘𝑧))(𝑁‘𝑧) ↔ (ℎ𝑀𝑤)((ℎ(1st ‘𝑍)𝑤)(Inv‘𝑇)(ℎ(1st ‘𝐸)𝑤))(ℎ𝑁𝑤))) | 
| 310 | 309 | ralxp 5851 | . . . . 5
⊢
(∀𝑧 ∈
((𝑂 Func 𝑆) × 𝐵)(𝑀‘𝑧)(((1st ‘𝑍)‘𝑧)(Inv‘𝑇)((1st ‘𝐸)‘𝑧))(𝑁‘𝑧) ↔ ∀ℎ ∈ (𝑂 Func 𝑆)∀𝑤 ∈ 𝐵 (ℎ𝑀𝑤)((ℎ(1st ‘𝑍)𝑤)(Inv‘𝑇)(ℎ(1st ‘𝐸)𝑤))(ℎ𝑁𝑤)) | 
| 311 | 295, 310 | sylibr 234 | . . . 4
⊢ (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀‘𝑧)(((1st ‘𝑍)‘𝑧)(Inv‘𝑇)((1st ‘𝐸)‘𝑧))(𝑁‘𝑧)) | 
| 312 | 311 | r19.21bi 3250 | . . 3
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)) → (𝑀‘𝑧)(((1st ‘𝑍)‘𝑧)(Inv‘𝑇)((1st ‘𝐸)‘𝑧))(𝑁‘𝑧)) | 
| 313 | 1, 8, 9, 22, 23, 24, 25, 27, 312 | invfuc 18023 | . 2
⊢ (𝜑 → 𝑀(𝑍𝐼𝐸)(𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ↦ (𝑁‘𝑧))) | 
| 314 |  | fvex 6918 | . . . . 5
⊢
((1st ‘𝑓)‘𝑥) ∈ V | 
| 315 | 314 | mptex 7244 | . . . 4
⊢ (𝑢 ∈ ((1st
‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))) ∈ V | 
| 316 | 42, 315 | fnmpoi 8096 | . . 3
⊢ 𝑁 Fn ((𝑂 Func 𝑆) × 𝐵) | 
| 317 |  | dffn5 6966 | . . 3
⊢ (𝑁 Fn ((𝑂 Func 𝑆) × 𝐵) ↔ 𝑁 = (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ↦ (𝑁‘𝑧))) | 
| 318 | 316, 317 | mpbi 230 | . 2
⊢ 𝑁 = (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ↦ (𝑁‘𝑧)) | 
| 319 | 313, 318 | breqtrrdi 5184 | 1
⊢ (𝜑 → 𝑀(𝑍𝐼𝐸)𝑁) |