| Step | Hyp | Ref
| Expression |
| 1 | | relfunc 17907 |
. . 3
⊢ Rel
(𝐶 Func 𝑄) |
| 2 | | yoneda.y |
. . . 4
⊢ 𝑌 = (Yon‘𝐶) |
| 3 | | yoneda.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ Cat) |
| 4 | | yoneda.o |
. . . 4
⊢ 𝑂 = (oppCat‘𝐶) |
| 5 | | yoneda.s |
. . . 4
⊢ 𝑆 = (SetCat‘𝑈) |
| 6 | | yoneda.q |
. . . 4
⊢ 𝑄 = (𝑂 FuncCat 𝑆) |
| 7 | | yoneda.w |
. . . . 5
⊢ (𝜑 → 𝑉 ∈ 𝑊) |
| 8 | | yoneda.v |
. . . . . 6
⊢ (𝜑 → (ran
(Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
| 9 | 8 | unssbd 4194 |
. . . . 5
⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
| 10 | 7, 9 | ssexd 5324 |
. . . 4
⊢ (𝜑 → 𝑈 ∈ V) |
| 11 | | yoneda.u |
. . . 4
⊢ (𝜑 → ran
(Homf ‘𝐶) ⊆ 𝑈) |
| 12 | 2, 3, 4, 5, 6, 10,
11 | yoncl 18307 |
. . 3
⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄)) |
| 13 | | 1st2nd 8064 |
. . 3
⊢ ((Rel
(𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = 〈(1st ‘𝑌), (2nd ‘𝑌)〉) |
| 14 | 1, 12, 13 | sylancr 587 |
. 2
⊢ (𝜑 → 𝑌 = 〈(1st ‘𝑌), (2nd ‘𝑌)〉) |
| 15 | | 1st2ndbr 8067 |
. . . . 5
⊢ ((Rel
(𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) |
| 16 | 1, 12, 15 | sylancr 587 |
. . . 4
⊢ (𝜑 → (1st
‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) |
| 17 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → (𝑁‘𝑣) = (𝑁‘〈((1st ‘𝑌)‘𝑤), 𝑧〉)) |
| 18 | | df-ov 7434 |
. . . . . . . . . . 11
⊢
(((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑁‘〈((1st ‘𝑌)‘𝑤), 𝑧〉) |
| 19 | 17, 18 | eqtr4di 2795 |
. . . . . . . . . 10
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → (𝑁‘𝑣) = (((1st ‘𝑌)‘𝑤)𝑁𝑧)) |
| 20 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → ((1st ‘𝐸)‘𝑣) = ((1st ‘𝐸)‘〈((1st ‘𝑌)‘𝑤), 𝑧〉)) |
| 21 | | df-ov 7434 |
. . . . . . . . . . . 12
⊢
(((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = ((1st ‘𝐸)‘〈((1st ‘𝑌)‘𝑤), 𝑧〉) |
| 22 | 20, 21 | eqtr4di 2795 |
. . . . . . . . . . 11
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → ((1st ‘𝐸)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)) |
| 23 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → ((1st ‘𝑍)‘𝑣) = ((1st ‘𝑍)‘〈((1st ‘𝑌)‘𝑤), 𝑧〉)) |
| 24 | | df-ov 7434 |
. . . . . . . . . . . 12
⊢
(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧) = ((1st ‘𝑍)‘〈((1st ‘𝑌)‘𝑤), 𝑧〉) |
| 25 | 23, 24 | eqtr4di 2795 |
. . . . . . . . . . 11
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → ((1st ‘𝑍)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧)) |
| 26 | 22, 25 | oveq12d 7449 |
. . . . . . . . . 10
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)) = ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧))) |
| 27 | 19, 26 | eleq12d 2835 |
. . . . . . . . 9
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → ((𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)) ↔ (((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧)))) |
| 28 | | yoneda.r |
. . . . . . . . . . . . . 14
⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇) |
| 29 | 28 | fucbas 18008 |
. . . . . . . . . . . . 13
⊢ ((𝑄 ×c
𝑂) Func 𝑇) = (Base‘𝑅) |
| 30 | | yonedainv.i |
. . . . . . . . . . . . 13
⊢ 𝐼 = (Inv‘𝑅) |
| 31 | | yoneda.b |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐵 = (Base‘𝐶) |
| 32 | | yoneda.1 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 =
(Id‘𝐶) |
| 33 | | yoneda.t |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑇 = (SetCat‘𝑉) |
| 34 | | yoneda.h |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐻 =
(HomF‘𝑄) |
| 35 | | yoneda.e |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐸 = (𝑂 evalF 𝑆) |
| 36 | | yoneda.z |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑍 = (𝐻 ∘func
((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉
∘func (𝑄 2ndF 𝑂))
〈,〉F (𝑄 1stF 𝑂))) |
| 37 | 2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 3, 7, 11, 8 | yonedalem1 18317 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) |
| 38 | 37 | simpld 494 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
| 39 | | funcrcl 17908 |
. . . . . . . . . . . . . . . 16
⊢ (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat)) |
| 40 | 38, 39 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat)) |
| 41 | 40 | simpld 494 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat) |
| 42 | 40 | simprd 495 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑇 ∈ Cat) |
| 43 | 28, 41, 42 | fuccat 18018 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑅 ∈ Cat) |
| 44 | 37 | simprd 495 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
| 45 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(Iso‘𝑅) =
(Iso‘𝑅) |
| 46 | | yoneda.m |
. . . . . . . . . . . . . 14
⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥)))) |
| 47 | | yonedainv.n |
. . . . . . . . . . . . . 14
⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))))) |
| 48 | 2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 3, 7, 11, 8,
46, 30, 47 | yonedainv 18326 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀(𝑍𝐼𝐸)𝑁) |
| 49 | 29, 30, 43, 38, 44, 45, 48 | inviso2 17811 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ (𝐸(Iso‘𝑅)𝑍)) |
| 50 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ (𝑄 ×c
𝑂) = (𝑄 ×c 𝑂) |
| 51 | 6 | fucbas 18008 |
. . . . . . . . . . . . . 14
⊢ (𝑂 Func 𝑆) = (Base‘𝑄) |
| 52 | 4, 31 | oppcbas 17761 |
. . . . . . . . . . . . . 14
⊢ 𝐵 = (Base‘𝑂) |
| 53 | 50, 51, 52 | xpcbas 18223 |
. . . . . . . . . . . . 13
⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂)) |
| 54 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ ((𝑄 ×c
𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇) |
| 55 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(Iso‘𝑇) =
(Iso‘𝑇) |
| 56 | 28, 53, 54, 44, 38, 45, 55 | fuciso 18023 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁 ∈ (𝐸(Iso‘𝑅)𝑍) ↔ (𝑁 ∈ (𝐸((𝑄 ×c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣))))) |
| 57 | 49, 56 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁 ∈ (𝐸((𝑄 ×c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)))) |
| 58 | 57 | simprd 495 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣))) |
| 59 | 58 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣))) |
| 60 | 31, 51, 16 | funcf1 17911 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘𝑌):𝐵⟶(𝑂 Func 𝑆)) |
| 61 | 60 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆)) |
| 62 | | simprr 773 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵) |
| 63 | 61, 62 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) |
| 64 | | simprl 771 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝐵) |
| 65 | 63, 64 | opelxpd 5724 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 〈((1st
‘𝑌)‘𝑤), 𝑧〉 ∈ ((𝑂 Func 𝑆) × 𝐵)) |
| 66 | 27, 59, 65 | rspcdva 3623 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧))) |
| 67 | 4 | oppccat 17765 |
. . . . . . . . . . . . 13
⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
| 68 | 3, 67 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑂 ∈ Cat) |
| 69 | 68 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑂 ∈ Cat) |
| 70 | 5 | setccat 18130 |
. . . . . . . . . . . . 13
⊢ (𝑈 ∈ V → 𝑆 ∈ Cat) |
| 71 | 10, 70 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ∈ Cat) |
| 72 | 71 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑆 ∈ Cat) |
| 73 | 35, 69, 72, 52, 63, 64 | evlf1 18265 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = ((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧)) |
| 74 | 3 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝐶 ∈ Cat) |
| 75 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
| 76 | 2, 31, 74, 62, 75, 64 | yon11 18309 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤)) |
| 77 | 73, 76 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = (𝑧(Hom ‘𝐶)𝑤)) |
| 78 | 7 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑉 ∈ 𝑊) |
| 79 | 11 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Homf
‘𝐶) ⊆ 𝑈) |
| 80 | 8 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ran (Homf
‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
| 81 | 2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 74, 78, 79, 80, 63, 64 | yonedalem21 18318 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧) = (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
| 82 | 77, 81 | oveq12d 7449 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧)) = ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))) |
| 83 | 66, 82 | eleqtrd 2843 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))) |
| 84 | 9 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ⊆ 𝑉) |
| 85 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(Base‘𝑆) =
(Base‘𝑆) |
| 86 | | relfunc 17907 |
. . . . . . . . . . . . . 14
⊢ Rel
(𝑂 Func 𝑆) |
| 87 | | 1st2ndbr 8067 |
. . . . . . . . . . . . . 14
⊢ ((Rel
(𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (1st
‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑤))) |
| 88 | 86, 63, 87 | sylancr 587 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st
‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑤))) |
| 89 | 52, 85, 88 | funcf1 17911 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st
‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆)) |
| 90 | 89, 64 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ (Base‘𝑆)) |
| 91 | 5, 10 | setcbas 18123 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑈 = (Base‘𝑆)) |
| 92 | 91 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 = (Base‘𝑆)) |
| 93 | 90, 92 | eleqtrrd 2844 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ 𝑈) |
| 94 | 76, 93 | eqeltrrd 2842 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑈) |
| 95 | 84, 94 | sseldd 3984 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑉) |
| 96 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Homf ‘𝑄) = (Homf ‘𝑄) |
| 97 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆) |
| 98 | 6, 97 | fuchom 18009 |
. . . . . . . . . 10
⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄) |
| 99 | 61, 64 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) |
| 100 | 96, 51, 98, 99, 63 | homfval 17735 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) = (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
| 101 | 8 | unssad 4193 |
. . . . . . . . . . 11
⊢ (𝜑 → ran
(Homf ‘𝑄) ⊆ 𝑉) |
| 102 | 101 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Homf
‘𝑄) ⊆ 𝑉) |
| 103 | 96, 51 | homffn 17736 |
. . . . . . . . . . 11
⊢
(Homf ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)) |
| 104 | | fnovrn 7608 |
. . . . . . . . . . 11
⊢
(((Homf ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)) ∧ ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ ran (Homf
‘𝑄)) |
| 105 | 103, 99, 63, 104 | mp3an2i 1468 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ ran (Homf
‘𝑄)) |
| 106 | 102, 105 | sseldd 3984 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ 𝑉) |
| 107 | 100, 106 | eqeltrrd 2842 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) ∈ 𝑉) |
| 108 | 33, 78, 95, 107, 55 | setciso 18136 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) ↔ (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))) |
| 109 | 83, 108 | mpbid 232 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
| 110 | 74 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat) |
| 111 | 110 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat) |
| 112 | 64 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧 ∈ 𝐵) |
| 113 | 112 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑧 ∈ 𝐵) |
| 114 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) |
| 115 | 2, 31, 111, 113, 75, 114 | yon11 18309 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦) = (𝑦(Hom ‘𝐶)𝑧)) |
| 116 | 115 | eqcomd 2743 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑦(Hom ‘𝐶)𝑧) = ((1st ‘((1st
‘𝑌)‘𝑧))‘𝑦)) |
| 117 | 111 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝐶 ∈ Cat) |
| 118 | 62 | ad3antrrr 730 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑤 ∈ 𝐵) |
| 119 | 113 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑧 ∈ 𝐵) |
| 120 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(comp‘𝐶) =
(comp‘𝐶) |
| 121 | 114 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑦 ∈ 𝐵) |
| 122 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) |
| 123 | | simpllr 776 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) |
| 124 | 2, 31, 117, 118, 75, 119, 120, 121, 122, 123 | yon12 18310 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ) = (ℎ(〈𝑦, 𝑧〉(comp‘𝐶)𝑤)𝑔)) |
| 125 | 2, 31, 117, 119, 75, 118, 120, 121, 123, 122 | yon2 18311 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔) = (ℎ(〈𝑦, 𝑧〉(comp‘𝐶)𝑤)𝑔)) |
| 126 | 124, 125 | eqtr4d 2780 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ) = ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔)) |
| 127 | 116, 126 | mpteq12dva 5231 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ)) = (𝑔 ∈ ((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔))) |
| 128 | 16 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) |
| 129 | 31, 75, 98, 128, 64, 62 | funcf2 17913 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
| 130 | 129 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) ∈ (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
| 131 | 97, 130 | nat1st2nd 17999 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) ∈ (〈(1st
‘((1st ‘𝑌)‘𝑧)), (2nd ‘((1st
‘𝑌)‘𝑧))〉(𝑂 Nat 𝑆)〈(1st
‘((1st ‘𝑌)‘𝑤)), (2nd ‘((1st
‘𝑌)‘𝑤))〉)) |
| 132 | 131 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) ∈ (〈(1st
‘((1st ‘𝑌)‘𝑧)), (2nd ‘((1st
‘𝑌)‘𝑧))〉(𝑂 Nat 𝑆)〈(1st
‘((1st ‘𝑌)‘𝑤)), (2nd ‘((1st
‘𝑌)‘𝑤))〉)) |
| 133 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢ (Hom
‘𝑆) = (Hom
‘𝑆) |
| 134 | 97, 132, 52, 133, 114 | natcl 18001 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦) ∈ (((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑤))‘𝑦))) |
| 135 | 10 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ∈ V) |
| 136 | 135 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 ∈ V) |
| 137 | 60 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆)) |
| 138 | 137, 112 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) |
| 139 | | 1st2ndbr 8067 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((Rel
(𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) → (1st
‘((1st ‘𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑧))) |
| 140 | 86, 138, 139 | sylancr 587 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st
‘((1st ‘𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑧))) |
| 141 | 52, 85, 140 | funcf1 17911 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st
‘((1st ‘𝑌)‘𝑧)):𝐵⟶(Base‘𝑆)) |
| 142 | 141 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦) ∈ (Base‘𝑆)) |
| 143 | 92 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 = (Base‘𝑆)) |
| 144 | 142, 143 | eleqtrrd 2844 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦) ∈ 𝑈) |
| 145 | 89 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st
‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆)) |
| 146 | 145 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑦) ∈ (Base‘𝑆)) |
| 147 | 146, 143 | eleqtrrd 2844 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑦) ∈ 𝑈) |
| 148 | 5, 136, 133, 144, 147 | elsetchom 18126 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦) ∈ (((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑤))‘𝑦)) ↔ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦):((1st ‘((1st
‘𝑌)‘𝑧))‘𝑦)⟶((1st
‘((1st ‘𝑌)‘𝑤))‘𝑦))) |
| 149 | 134, 148 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦):((1st ‘((1st
‘𝑌)‘𝑧))‘𝑦)⟶((1st
‘((1st ‘𝑌)‘𝑤))‘𝑦)) |
| 150 | 149 | feqmptd 6977 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦) = (𝑔 ∈ ((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔))) |
| 151 | 127, 150 | eqtr4d 2780 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ)) = (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)) |
| 152 | 151 | mpteq2dva 5242 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ))) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦))) |
| 153 | 78 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉 ∈ 𝑊) |
| 154 | 79 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Homf
‘𝐶) ⊆ 𝑈) |
| 155 | 80 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Homf
‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
| 156 | 63 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) |
| 157 | 76 | eleq2d 2827 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ℎ ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ↔ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤))) |
| 158 | 157 | biimpar 477 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ℎ ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧)) |
| 159 | 2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 110, 153, 154, 155, 156, 112, 47, 158 | yonedalem4a 18320 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ)))) |
| 160 | 97, 131, 52 | natfn 18002 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵) |
| 161 | | dffn5 6967 |
. . . . . . . . . . 11
⊢ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵 ↔ ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦))) |
| 162 | 160, 161 | sylib 218 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦))) |
| 163 | 152, 159,
162 | 3eqtr4d 2787 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ) = ((𝑧(2nd ‘𝑌)𝑤)‘ℎ)) |
| 164 | 163 | mpteq2dva 5242 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ)) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd ‘𝑌)𝑤)‘ℎ))) |
| 165 | | f1of 6848 |
. . . . . . . . . 10
⊢
((((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
| 166 | 109, 165 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
| 167 | 166 | feqmptd 6977 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ))) |
| 168 | 129 | feqmptd 6977 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2nd ‘𝑌)𝑤) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd ‘𝑌)𝑤)‘ℎ))) |
| 169 | 164, 167,
168 | 3eqtr4d 2787 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd ‘𝑌)𝑤)) |
| 170 | 169 | f1oeq1d 6843 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) ↔ (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))) |
| 171 | 109, 170 | mpbid 232 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
| 172 | 171 | ralrimivva 3202 |
. . . 4
⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
| 173 | 31, 75, 98 | isffth2 17963 |
. . . 4
⊢
((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ ((1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))) |
| 174 | 16, 172, 173 | sylanbrc 583 |
. . 3
⊢ (𝜑 → (1st
‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌)) |
| 175 | | df-br 5144 |
. . 3
⊢
((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ 〈(1st ‘𝑌), (2nd ‘𝑌)〉 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))) |
| 176 | 174, 175 | sylib 218 |
. 2
⊢ (𝜑 → 〈(1st
‘𝑌), (2nd
‘𝑌)〉 ∈
((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))) |
| 177 | 14, 176 | eqeltrd 2841 |
1
⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))) |