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Theorem yonedainv 18327
Description: The Yoneda Lemma with explicit inverse. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y 𝑌 = (Yon‘𝐶)
yoneda.b 𝐵 = (Base‘𝐶)
yoneda.1 1 = (Id‘𝐶)
yoneda.o 𝑂 = (oppCat‘𝐶)
yoneda.s 𝑆 = (SetCat‘𝑈)
yoneda.t 𝑇 = (SetCat‘𝑉)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomF𝑄)
yoneda.r 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (𝜑𝐶 ∈ Cat)
yoneda.w (𝜑𝑉𝑊)
yoneda.u (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
yoneda.v (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
yoneda.m 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
yonedainv.i 𝐼 = (Inv‘𝑅)
yonedainv.n 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
Assertion
Ref Expression
yonedainv (𝜑𝑀(𝑍𝐼𝐸)𝑁)
Distinct variable groups:   𝑓,𝑎,𝑔,𝑥,𝑦, 1   𝑢,𝑎,𝑔,𝑦,𝐶,𝑓,𝑥   𝐸,𝑎,𝑓,𝑔,𝑢,𝑦   𝐵,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑁,𝑎   𝑂,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑆,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑔,𝑀,𝑢,𝑦   𝑄,𝑎,𝑓,𝑔,𝑢,𝑥   𝑇,𝑓,𝑔,𝑢,𝑦   𝜑,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑢,𝑅   𝑌,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑍,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦
Allowed substitution hints:   𝑄(𝑦)   𝑅(𝑥,𝑦,𝑓,𝑔,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   1 (𝑢)   𝐸(𝑥)   𝐻(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝐼(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑁(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝑊(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)

Proof of Theorem yonedainv
Dummy variables 𝑏 𝑘 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef 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)
43fucbas 18009 . . . 4 (𝑂 Func 𝑆) = (Base‘𝑄)
5 yoneda.o . . . . 5 𝑂 = (oppCat‘𝐶)
6 yoneda.b . . . . 5 𝐵 = (Base‘𝐶)
75, 6oppcbas 17762 . . . 4 𝐵 = (Base‘𝑂)
82, 4, 7xpcbas 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 = (HomF𝑄)
15 yoneda.e . . . . 5 𝐸 = (𝑂 evalF 𝑆)
16 yoneda.z . . . . 5 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
17 yoneda.c . . . . 5 (𝜑𝐶 ∈ Cat)
18 yoneda.w . . . . 5 (𝜑𝑉𝑊)
19 yoneda.u . . . . 5 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
20 yoneda.v . . . . 5 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
2110, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 17, 18, 19, 20yonedalem1 18318 . . . 4 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
2221simpld 494 . . 3 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
2321simprd 495 . . 3 (𝜑𝐸 ∈ ((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), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
2710, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 17, 18, 19, 20, 26yonedalem3 18326 . . 3 (𝜑𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))
2817adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → 𝐶 ∈ Cat)
2918adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → 𝑉𝑊)
3019adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ran (Homf𝐶) ⊆ 𝑈)
3120adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
32 simprl 770 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ∈ (𝑂 Func 𝑆))
33 simprr 772 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → 𝑤𝐵)
3410, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 28, 29, 30, 31, 32, 33, 26yonedalem3a 18320 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))) ∧ (𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤)))
3534simprd 495 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤))
3628adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → 𝐶 ∈ Cat)
3729adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → 𝑉𝑊)
3830adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → ran (Homf𝐶) ⊆ 𝑈)
3931adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
40 simplrl 776 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → ∈ (𝑂 Func 𝑆))
41 simplrr 777 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → 𝑤𝐵)
42 yonedainv.n . . . . . . . . . . . 12 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
43 simpr 484 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → 𝑏 ∈ ((1st)‘𝑤))
4410, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 36, 37, 38, 39, 40, 41, 42, 43yonedalem4c 18323 . . . . . . . . . . 11 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st)‘𝑤)) → ((𝑁𝑤)‘𝑏) ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
4544fmpttd 7134 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑏 ∈ ((1st)‘𝑤) ↦ ((𝑁𝑤)‘𝑏)):((1st)‘𝑤)⟶(((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
466fvexi 6919 . . . . . . . . . . . . . . 15 𝐵 ∈ V
4746mptex 7244 . . . . . . . . . . . . . 14 (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢))) ∈ V
48 eqid 2736 . . . . . . . . . . . . . 14 (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))) = (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢))))
4947, 48fnmpti 6710 . . . . . . . . . . . . 13 (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))) Fn ((1st)‘𝑤)
50 simpl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑓 = 𝑥 = 𝑤) → 𝑓 = )
5150fveq2d 6909 . . . . . . . . . . . . . . . . . 18 ((𝑓 = 𝑥 = 𝑤) → (1st𝑓) = (1st))
52 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝑓 = 𝑥 = 𝑤) → 𝑥 = 𝑤)
5351, 52fveq12d 6912 . . . . . . . . . . . . . . . . 17 ((𝑓 = 𝑥 = 𝑤) → ((1st𝑓)‘𝑥) = ((1st)‘𝑤))
54 simplr 768 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → 𝑥 = 𝑤)
5554oveq2d 7448 . . . . . . . . . . . . . . . . . . 19 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐶)𝑤))
56 simpll 766 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → 𝑓 = )
5756fveq2d 6909 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → (2nd𝑓) = (2nd))
58 eqidd 2737 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → 𝑦 = 𝑦)
5957, 54, 58oveq123d 7453 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → (𝑥(2nd𝑓)𝑦) = (𝑤(2nd)𝑦))
6059fveq1d 6907 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → ((𝑥(2nd𝑓)𝑦)‘𝑔) = ((𝑤(2nd)𝑦)‘𝑔))
6160fveq1d 6907 . . . . . . . . . . . . . . . . . . 19 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢) = (((𝑤(2nd)𝑦)‘𝑔)‘𝑢))
6255, 61mpteq12dv 5232 . . . . . . . . . . . . . . . . . 18 (((𝑓 = 𝑥 = 𝑤) ∧ 𝑦𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))
6362mpteq2dva 5241 . . . . . . . . . . . . . . . . 17 ((𝑓 = 𝑥 = 𝑤) → (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢))) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢))))
6453, 63mpteq12dv 5232 . . . . . . . . . . . . . . . 16 ((𝑓 = 𝑥 = 𝑤) → (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))) = (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))))
65 fvex 6918 . . . . . . . . . . . . . . . . 17 ((1st)‘𝑤) ∈ V
6665mptex 7244 . . . . . . . . . . . . . . . 16 (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))) ∈ V
6764, 42, 66ovmpoa 7589 . . . . . . . . . . . . . . 15 (( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵) → (𝑁𝑤) = (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))))
6867adantl 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑁𝑤) = (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))))
6968fneq1d 6660 . . . . . . . . . . . . 13 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑁𝑤) Fn ((1st)‘𝑤) ↔ (𝑢 ∈ ((1st)‘𝑤) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑤) ↦ (((𝑤(2nd)𝑦)‘𝑔)‘𝑢)))) Fn ((1st)‘𝑤)))
7049, 69mpbiri 258 . . . . . . . . . . . 12 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑁𝑤) Fn ((1st)‘𝑤))
71 dffn5 6966 . . . . . . . . . . . 12 ((𝑁𝑤) Fn ((1st)‘𝑤) ↔ (𝑁𝑤) = (𝑏 ∈ ((1st)‘𝑤) ↦ ((𝑁𝑤)‘𝑏)))
7270, 71sylib 218 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑁𝑤) = (𝑏 ∈ ((1st)‘𝑤) ↦ ((𝑁𝑤)‘𝑏)))
735oppccat 17766 . . . . . . . . . . . . . 14 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
7417, 73syl 17 . . . . . . . . . . . . 13 (𝜑𝑂 ∈ Cat)
7574adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → 𝑂 ∈ Cat)
7620unssbd 4193 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
7718, 76ssexd 5323 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ V)
7812setccat 18131 . . . . . . . . . . . . . 14 (𝑈 ∈ V → 𝑆 ∈ Cat)
7977, 78syl 17 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ Cat)
8079adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → 𝑆 ∈ Cat)
8115, 75, 80, 7, 32, 33evlf1 18266 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((1st𝐸)𝑤) = ((1st)‘𝑤))
8210, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 28, 29, 30, 31, 32, 33yonedalem21 18319 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((1st𝑍)𝑤) = (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
8372, 81, 82feq123d 6724 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑁𝑤):((1st𝐸)𝑤)⟶((1st𝑍)𝑤) ↔ (𝑏 ∈ ((1st)‘𝑤) ↦ ((𝑁𝑤)‘𝑏)):((1st)‘𝑤)⟶(((1st𝑌)‘𝑤)(𝑂 Nat 𝑆))))
8445, 83mpbird 257 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑁𝑤):((1st𝐸)𝑤)⟶((1st𝑍)𝑤))
85 fcompt 7152 . . . . . . . . . . 11 (((𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤) ∧ (𝑁𝑤):((1st𝐸)𝑤)⟶((1st𝑍)𝑤)) → ((𝑀𝑤) ∘ (𝑁𝑤)) = (𝑘 ∈ ((1st𝐸)𝑤) ↦ ((𝑀𝑤)‘((𝑁𝑤)‘𝑘))))
8635, 84, 85syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤) ∘ (𝑁𝑤)) = (𝑘 ∈ ((1st𝐸)𝑤) ↦ ((𝑀𝑤)‘((𝑁𝑤)‘𝑘))))
8781eleq2d 2826 . . . . . . . . . . . . . 14 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑘 ∈ ((1st𝐸)𝑤) ↔ 𝑘 ∈ ((1st)‘𝑤)))
8887biimpa 476 . . . . . . . . . . . . 13 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st𝐸)𝑤)) → 𝑘 ∈ ((1st)‘𝑤))
8928adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝐶 ∈ Cat)
9029adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝑉𝑊)
9130adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ran (Homf𝐶) ⊆ 𝑈)
9231adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
93 simplrl 776 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ∈ (𝑂 Func 𝑆))
94 simplrr 777 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝑤𝐵)
9510, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 89, 90, 91, 92, 93, 94, 26yonedalem3a 18320 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑀𝑤) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))) ∧ (𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤)))
9695simpld 494 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (𝑀𝑤) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))))
9796fveq1d 6907 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑀𝑤)‘((𝑁𝑤)‘𝑘)) = ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘((𝑁𝑤)‘𝑘)))
9872, 44fmpt3d 7135 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑁𝑤):((1st)‘𝑤)⟶(((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
9998ffvelcdmda 7103 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑁𝑤)‘𝑘) ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
100 fveq1 6904 . . . . . . . . . . . . . . . . 17 (𝑎 = ((𝑁𝑤)‘𝑘) → (𝑎𝑤) = (((𝑁𝑤)‘𝑘)‘𝑤))
101100fveq1d 6907 . . . . . . . . . . . . . . . 16 (𝑎 = ((𝑁𝑤)‘𝑘) → ((𝑎𝑤)‘( 1𝑤)) = ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)))
102 eqid 2736 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))
103 fvex 6918 . . . . . . . . . . . . . . . 16 ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)) ∈ V
104101, 102, 103fvmpt 7015 . . . . . . . . . . . . . . 15 (((𝑁𝑤)‘𝑘) ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) → ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘((𝑁𝑤)‘𝑘)) = ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)))
10599, 104syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘((𝑁𝑤)‘𝑘)) = ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)))
106 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝑘 ∈ ((1st)‘𝑤))
107 eqid 2736 . . . . . . . . . . . . . . . . 17 (Hom ‘𝐶) = (Hom ‘𝐶)
1086, 107, 11, 89, 94catidcl 17726 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ( 1𝑤) ∈ (𝑤(Hom ‘𝐶)𝑤))
10910, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 89, 90, 91, 92, 93, 94, 42, 106, 94, 108yonedalem4b 18322 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)) = (((𝑤(2nd)𝑤)‘( 1𝑤))‘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) ∧ ∈ (𝑂 Func 𝑆)) → (1st)(𝑂 Func 𝑆)(2nd))
114112, 93, 113sylancr 587 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (1st)(𝑂 Func 𝑆)(2nd))
1157, 110, 111, 114, 94funcid 17916 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑤(2nd)𝑤)‘((Id‘𝑂)‘𝑤)) = ((Id‘𝑆)‘((1st)‘𝑤)))
1165, 11oppcid 17765 . . . . . . . . . . . . . . . . . . . 20 (𝐶 ∈ Cat → (Id‘𝑂) = 1 )
11789, 116syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (Id‘𝑂) = 1 )
118117fveq1d 6907 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((Id‘𝑂)‘𝑤) = ( 1𝑤))
119118fveq2d 6909 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑤(2nd)𝑤)‘((Id‘𝑂)‘𝑤)) = ((𝑤(2nd)𝑤)‘( 1𝑤)))
12077ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝑈 ∈ V)
121 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 (Base‘𝑆) = (Base‘𝑆)
1227, 121, 114funcf1 17912 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (1st):𝐵⟶(Base‘𝑆))
12312, 120setcbas 18124 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → 𝑈 = (Base‘𝑆))
124123feq3d 6722 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((1st):𝐵𝑈 ↔ (1st):𝐵⟶(Base‘𝑆)))
125122, 124mpbird 257 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (1st):𝐵𝑈)
126125, 94ffvelcdmd 7104 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((1st)‘𝑤) ∈ 𝑈)
12712, 111, 120, 126setcid 18132 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((Id‘𝑆)‘((1st)‘𝑤)) = ( I ↾ ((1st)‘𝑤)))
128115, 119, 1273eqtr3d 2784 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑤(2nd)𝑤)‘( 1𝑤)) = ( I ↾ ((1st)‘𝑤)))
129128fveq1d 6907 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (((𝑤(2nd)𝑤)‘( 1𝑤))‘𝑘) = (( I ↾ ((1st)‘𝑤))‘𝑘))
130 fvresi 7194 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ((1st)‘𝑤) → (( I ↾ ((1st)‘𝑤))‘𝑘) = 𝑘)
131130adantl 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → (( I ↾ ((1st)‘𝑤))‘𝑘) = 𝑘)
132109, 129, 1313eqtrd 2780 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((((𝑁𝑤)‘𝑘)‘𝑤)‘( 1𝑤)) = 𝑘)
13397, 105, 1323eqtrd 2780 . . . . . . . . . . . . 13 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st)‘𝑤)) → ((𝑀𝑤)‘((𝑁𝑤)‘𝑘)) = 𝑘)
13488, 133syldan 591 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑘 ∈ ((1st𝐸)𝑤)) → ((𝑀𝑤)‘((𝑁𝑤)‘𝑘)) = 𝑘)
135134mpteq2dva 5241 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑘 ∈ ((1st𝐸)𝑤) ↦ ((𝑀𝑤)‘((𝑁𝑤)‘𝑘))) = (𝑘 ∈ ((1st𝐸)𝑤) ↦ 𝑘))
136 mptresid 6068 . . . . . . . . . . 11 ( I ↾ ((1st𝐸)𝑤)) = (𝑘 ∈ ((1st𝐸)𝑤) ↦ 𝑘)
137135, 136eqtr4di 2794 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑘 ∈ ((1st𝐸)𝑤) ↦ ((𝑀𝑤)‘((𝑁𝑤)‘𝑘))) = ( I ↾ ((1st𝐸)𝑤)))
13886, 137eqtrd 2776 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤) ∘ (𝑁𝑤)) = ( I ↾ ((1st𝐸)𝑤)))
139 fcompt 7152 . . . . . . . . . . 11 (((𝑁𝑤):((1st𝐸)𝑤)⟶((1st𝑍)𝑤) ∧ (𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤)) → ((𝑁𝑤) ∘ (𝑀𝑤)) = (𝑏 ∈ ((1st𝑍)𝑤) ↦ ((𝑁𝑤)‘((𝑀𝑤)‘𝑏))))
14084, 35, 139syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑁𝑤) ∘ (𝑀𝑤)) = (𝑏 ∈ ((1st𝑍)𝑤) ↦ ((𝑁𝑤)‘((𝑀𝑤)‘𝑏))))
141 eqid 2736 . . . . . . . . . . . . . 14 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
14228adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝐶 ∈ Cat)
14329adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑉𝑊)
14430adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ran (Homf𝐶) ⊆ 𝑈)
14531adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
146 simplrl 776 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ∈ (𝑂 Func 𝑆))
147 simplrr 777 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑤𝐵)
14881feq3d 6722 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤) ↔ (𝑀𝑤):((1st𝑍)𝑤)⟶((1st)‘𝑤)))
14935, 148mpbid 232 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑀𝑤):((1st𝑍)𝑤)⟶((1st)‘𝑤))
150149ffvelcdmda 7103 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑀𝑤)‘𝑏) ∈ ((1st)‘𝑤))
15110, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 142, 143, 144, 145, 146, 147, 42, 150yonedalem4c 18323 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑁𝑤)‘((𝑀𝑤)‘𝑏)) ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
152141, 151nat1st2nd 18000 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑁𝑤)‘((𝑀𝑤)‘𝑏)) ∈ (⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩(𝑂 Nat 𝑆)⟨(1st), (2nd)⟩))
153141, 152, 7natfn 18003 . . . . . . . . . . . . 13 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑁𝑤)‘((𝑀𝑤)‘𝑏)) Fn 𝐵)
15482eleq2d 2826 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑏 ∈ ((1st𝑍)𝑤) ↔ 𝑏 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆))))
155154biimpa 476 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑏 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
156141, 155nat1st2nd 18000 . . . . . . . . . . . . . 14 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑏 ∈ (⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩(𝑂 Nat 𝑆)⟨(1st), (2nd)⟩))
157141, 156, 7natfn 18003 . . . . . . . . . . . . 13 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑏 Fn 𝐵)
158142adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → 𝐶 ∈ Cat)
159147adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → 𝑤𝐵)
160 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → 𝑧𝐵)
16110, 6, 158, 159, 107, 160yon11 18310 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤))
162161eleq2d 2826 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↔ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))
163162biimpa 476 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤))
164158adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat)
165143ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉𝑊)
166144ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Homf𝐶) ⊆ 𝑈)
167145ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
168146ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ∈ (𝑂 Func 𝑆))
169159adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑤𝐵)
170150ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑀𝑤)‘𝑏) ∈ ((1st)‘𝑤))
171 simplr 768 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧𝐵)
172 simpr 484 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤))
17310, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 164, 165, 166, 167, 168, 169, 42, 170, 171, 172yonedalem4b 18322 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧)‘𝑘) = (((𝑤(2nd)𝑧)‘𝑘)‘((𝑀𝑤)‘𝑏)))
17410, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 164, 165, 166, 167, 168, 169, 26yonedalem3a 18320 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑀𝑤) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))) ∧ (𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤)))
175174simpld 494 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑀𝑤) = (𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤))))
176175fveq1d 6907 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑀𝑤)‘𝑏) = ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘𝑏))
177155ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑏 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)))
178 fveq1 6904 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑏 → (𝑎𝑤) = (𝑏𝑤))
179178fveq1d 6907 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 𝑏 → ((𝑎𝑤)‘( 1𝑤)) = ((𝑏𝑤)‘( 1𝑤)))
180 fvex 6918 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏𝑤)‘( 1𝑤)) ∈ V
181179, 102, 180fvmpt 7015 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) → ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘𝑏) = ((𝑏𝑤)‘( 1𝑤)))
182177, 181syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑎 ∈ (((1st𝑌)‘𝑤)(𝑂 Nat 𝑆)) ↦ ((𝑎𝑤)‘( 1𝑤)))‘𝑏) = ((𝑏𝑤)‘( 1𝑤)))
183176, 182eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑀𝑤)‘𝑏) = ((𝑏𝑤)‘( 1𝑤)))
184183fveq2d 6909 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd)𝑧)‘𝑘)‘((𝑀𝑤)‘𝑏)) = (((𝑤(2nd)𝑧)‘𝑘)‘((𝑏𝑤)‘( 1𝑤))))
185156ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑏 ∈ (⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩(𝑂 Nat 𝑆)⟨(1st), (2nd)⟩))
186 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (Hom ‘𝑂) = (Hom ‘𝑂)
187 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (comp‘𝑆) = (comp‘𝑆)
188107, 5oppchom 17759 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑤)
189172, 188eleqtrrdi 2851 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑘 ∈ (𝑤(Hom ‘𝑂)𝑧))
190141, 185, 7, 186, 187, 169, 171, 189nati 18004 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏𝑧)(⟨((1st ‘((1st𝑌)‘𝑤))‘𝑤), ((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟩(comp‘𝑆)((1st)‘𝑧))((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)) = (((𝑤(2nd)𝑧)‘𝑘)(⟨((1st ‘((1st𝑌)‘𝑤))‘𝑤), ((1st)‘𝑤)⟩(comp‘𝑆)((1st)‘𝑧))(𝑏𝑤)))
19177ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑈 ∈ V)
192191adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → 𝑈 ∈ V)
193192adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑈 ∈ V)
194 relfunc 17908 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Rel (𝐶 Func 𝑄)
19510, 17, 5, 12, 3, 77, 19yoncl 18308 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
196 1st2ndbr 8068 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
197194, 195, 196sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
1986, 4, 197funcf1 17912 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
199198ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
200199, 147ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
201 1st2ndbr 8068 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
202112, 200, 201sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
2037, 121, 202funcf1 17912 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st ‘((1st𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
20412, 191setcbas 18124 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → 𝑈 = (Base‘𝑆))
205204feq3d 6722 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st ‘((1st𝑌)‘𝑤)):𝐵𝑈 ↔ (1st ‘((1st𝑌)‘𝑤)):𝐵⟶(Base‘𝑆)))
206203, 205mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st ‘((1st𝑌)‘𝑤)):𝐵𝑈)
207206, 147ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑤) ∈ 𝑈)
208207ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑤) ∈ 𝑈)
209206ffvelcdmda 7103 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
210209adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
211112, 146, 113sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st)(𝑂 Func 𝑆)(2nd))
2127, 121, 211funcf1 17912 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st):𝐵⟶(Base‘𝑆))
213204feq3d 6722 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st):𝐵𝑈 ↔ (1st):𝐵⟶(Base‘𝑆)))
214212, 213mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (1st):𝐵𝑈)
215214ffvelcdmda 7103 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((1st)‘𝑧) ∈ 𝑈)
216215adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st)‘𝑧) ∈ 𝑈)
217 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Hom ‘𝑆) = (Hom ‘𝑆)
218202ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
2197, 186, 217, 218, 169, 171funcf2 17914 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧):(𝑤(Hom ‘𝑂)𝑧)⟶(((1st ‘((1st𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑤))‘𝑧)))
220219, 189ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑤))‘𝑧)))
22112, 193, 217, 208, 210elsetchom 18127 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑤))‘𝑧)) ↔ ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘):((1st ‘((1st𝑌)‘𝑤))‘𝑤)⟶((1st ‘((1st𝑌)‘𝑤))‘𝑧)))
222220, 221mpbid 232 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘):((1st ‘((1st𝑌)‘𝑤))‘𝑤)⟶((1st ‘((1st𝑌)‘𝑤))‘𝑧))
223156adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → 𝑏 ∈ (⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩(𝑂 Nat 𝑆)⟨(1st), (2nd)⟩))
224141, 223, 7, 217, 160natcl 18002 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (𝑏𝑧) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st)‘𝑧)))
22512, 192, 217, 209, 215elsetchom 18127 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((𝑏𝑧) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st)‘𝑧)) ↔ (𝑏𝑧):((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟶((1st)‘𝑧)))
226224, 225mpbid 232 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (𝑏𝑧):((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟶((1st)‘𝑧))
227226adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑏𝑧):((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟶((1st)‘𝑧))
22812, 193, 187, 208, 210, 216, 222, 227setcco 18129 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏𝑧)(⟨((1st ‘((1st𝑌)‘𝑤))‘𝑤), ((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟩(comp‘𝑆)((1st)‘𝑧))((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)) = ((𝑏𝑧) ∘ ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)))
229214, 147ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st)‘𝑤) ∈ 𝑈)
230229ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st)‘𝑤) ∈ 𝑈)
231141, 156, 7, 217, 147natcl 18002 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (𝑏𝑤) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st)‘𝑤)))
23212, 191, 217, 207, 229elsetchom 18127 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑏𝑤) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑤)(Hom ‘𝑆)((1st)‘𝑤)) ↔ (𝑏𝑤):((1st ‘((1st𝑌)‘𝑤))‘𝑤)⟶((1st)‘𝑤)))
233231, 232mpbid 232 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → (𝑏𝑤):((1st ‘((1st𝑌)‘𝑤))‘𝑤)⟶((1st)‘𝑤))
234233ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑏𝑤):((1st ‘((1st𝑌)‘𝑤))‘𝑤)⟶((1st)‘𝑤))
235112, 168, 113sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st)(𝑂 Func 𝑆)(2nd))
2367, 186, 217, 235, 169, 171funcf2 17914 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑤(2nd)𝑧):(𝑤(Hom ‘𝑂)𝑧)⟶(((1st)‘𝑤)(Hom ‘𝑆)((1st)‘𝑧)))
237236, 189ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd)𝑧)‘𝑘) ∈ (((1st)‘𝑤)(Hom ‘𝑆)((1st)‘𝑧)))
23812, 193, 217, 230, 216elsetchom 18127 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd)𝑧)‘𝑘) ∈ (((1st)‘𝑤)(Hom ‘𝑆)((1st)‘𝑧)) ↔ ((𝑤(2nd)𝑧)‘𝑘):((1st)‘𝑤)⟶((1st)‘𝑧)))
239237, 238mpbid 232 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑤(2nd)𝑧)‘𝑘):((1st)‘𝑤)⟶((1st)‘𝑧))
24012, 193, 187, 208, 230, 216, 234, 239setcco 18129 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd)𝑧)‘𝑘)(⟨((1st ‘((1st𝑌)‘𝑤))‘𝑤), ((1st)‘𝑤)⟩(comp‘𝑆)((1st)‘𝑧))(𝑏𝑤)) = (((𝑤(2nd)𝑧)‘𝑘) ∘ (𝑏𝑤)))
241190, 228, 2403eqtr3d 2784 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏𝑧) ∘ ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)) = (((𝑤(2nd)𝑧)‘𝑘) ∘ (𝑏𝑤)))
242241fveq1d 6907 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑏𝑧) ∘ ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘))‘( 1𝑤)) = ((((𝑤(2nd)𝑧)‘𝑘) ∘ (𝑏𝑤))‘( 1𝑤)))
2436, 107, 11, 142, 147catidcl 17726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ( 1𝑤) ∈ (𝑤(Hom ‘𝐶)𝑤))
24410, 6, 142, 147, 107, 147yon11 18310 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑤) = (𝑤(Hom ‘𝐶)𝑤))
245243, 244eleqtrrd 2843 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ( 1𝑤) ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑤))
246245ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ( 1𝑤) ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑤))
247222, 246fvco3d 7008 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑏𝑧) ∘ ((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘))‘( 1𝑤)) = ((𝑏𝑧)‘(((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)‘( 1𝑤))))
248233, 245fvco3d 7008 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((((𝑤(2nd)𝑧)‘𝑘) ∘ (𝑏𝑤))‘( 1𝑤)) = (((𝑤(2nd)𝑧)‘𝑘)‘((𝑏𝑤)‘( 1𝑤))))
249248ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((𝑤(2nd)𝑧)‘𝑘) ∘ (𝑏𝑤))‘( 1𝑤)) = (((𝑤(2nd)𝑧)‘𝑘)‘((𝑏𝑤)‘( 1𝑤))))
250242, 247, 2493eqtr3d 2784 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏𝑧)‘(((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)‘( 1𝑤))) = (((𝑤(2nd)𝑧)‘𝑘)‘((𝑏𝑤)‘( 1𝑤))))
251 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 (comp‘𝐶) = (comp‘𝐶)
252243ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ( 1𝑤) ∈ (𝑤(Hom ‘𝐶)𝑤))
25310, 6, 164, 169, 107, 169, 251, 171, 172, 252yon12 18311 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)‘( 1𝑤)) = (( 1𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐶)𝑤)𝑘))
2546, 107, 11, 164, 171, 251, 169, 172catlid 17727 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (( 1𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐶)𝑤)𝑘) = 𝑘)
255253, 254eqtrd 2776 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)‘( 1𝑤)) = 𝑘)
256255fveq2d 6909 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑏𝑧)‘(((𝑤(2nd ‘((1st𝑌)‘𝑤))𝑧)‘𝑘)‘( 1𝑤))) = ((𝑏𝑧)‘𝑘))
257250, 256eqtr3d 2778 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((𝑤(2nd)𝑧)‘𝑘)‘((𝑏𝑤)‘( 1𝑤))) = ((𝑏𝑧)‘𝑘))
258173, 184, 2573eqtrd 2780 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧)‘𝑘) = ((𝑏𝑧)‘𝑘))
259163, 258syldan 591 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) ∧ 𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧)) → ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧)‘𝑘) = ((𝑏𝑧)‘𝑘))
260259mpteq2dva 5241 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↦ ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧)‘𝑘)) = (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↦ ((𝑏𝑧)‘𝑘)))
261152adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((𝑁𝑤)‘((𝑀𝑤)‘𝑏)) ∈ (⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩(𝑂 Nat 𝑆)⟨(1st), (2nd)⟩))
262141, 261, 7, 217, 160natcl 18002 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st)‘𝑧)))
26312, 192, 217, 209, 215elsetchom 18127 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧) ∈ (((1st ‘((1st𝑌)‘𝑤))‘𝑧)(Hom ‘𝑆)((1st)‘𝑧)) ↔ (((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧):((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟶((1st)‘𝑧)))
264262, 263mpbid 232 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧):((1st ‘((1st𝑌)‘𝑤))‘𝑧)⟶((1st)‘𝑧))
265264feqmptd 6976 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧) = (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↦ ((((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧)‘𝑘)))
266226feqmptd 6976 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (𝑏𝑧) = (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↦ ((𝑏𝑧)‘𝑘)))
267260, 265, 2663eqtr4d 2786 . . . . . . . . . . . . 13 ((((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) ∧ 𝑧𝐵) → (((𝑁𝑤)‘((𝑀𝑤)‘𝑏))‘𝑧) = (𝑏𝑧))
268153, 157, 267eqfnfvd 7053 . . . . . . . . . . . 12 (((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) ∧ 𝑏 ∈ ((1st𝑍)𝑤)) → ((𝑁𝑤)‘((𝑀𝑤)‘𝑏)) = 𝑏)
269268mpteq2dva 5241 . . . . . . . . . . 11 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑏 ∈ ((1st𝑍)𝑤) ↦ ((𝑁𝑤)‘((𝑀𝑤)‘𝑏))) = (𝑏 ∈ ((1st𝑍)𝑤) ↦ 𝑏))
270 mptresid 6068 . . . . . . . . . . 11 ( I ↾ ((1st𝑍)𝑤)) = (𝑏 ∈ ((1st𝑍)𝑤) ↦ 𝑏)
271269, 270eqtr4di 2794 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑏 ∈ ((1st𝑍)𝑤) ↦ ((𝑁𝑤)‘((𝑀𝑤)‘𝑏))) = ( I ↾ ((1st𝑍)𝑤)))
272140, 271eqtrd 2776 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑁𝑤) ∘ (𝑀𝑤)) = ( I ↾ ((1st𝑍)𝑤)))
273 fcof1o 7317 . . . . . . . . 9 ((((𝑀𝑤):((1st𝑍)𝑤)⟶((1st𝐸)𝑤) ∧ (𝑁𝑤):((1st𝐸)𝑤)⟶((1st𝑍)𝑤)) ∧ (((𝑀𝑤) ∘ (𝑁𝑤)) = ( I ↾ ((1st𝐸)𝑤)) ∧ ((𝑁𝑤) ∘ (𝑀𝑤)) = ( I ↾ ((1st𝑍)𝑤)))) → ((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑀𝑤) = (𝑁𝑤)))
27435, 84, 138, 272, 273syl22anc 838 . . . . . . . 8 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑀𝑤) = (𝑁𝑤)))
275 eqcom 2743 . . . . . . . . 9 ((𝑀𝑤) = (𝑁𝑤) ↔ (𝑁𝑤) = (𝑀𝑤))
276275anbi2i 623 . . . . . . . 8 (((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑀𝑤) = (𝑁𝑤)) ↔ ((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑁𝑤) = (𝑀𝑤)))
277274, 276sylib 218 . . . . . . 7 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑁𝑤) = (𝑀𝑤)))
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)) → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
281279, 22, 280sylancr 587 . . . . . . . . . . 11 (𝜑 → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
2828, 278, 281funcf1 17912 . . . . . . . . . 10 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
28313, 18setcbas 18124 . . . . . . . . . . 11 (𝜑𝑉 = (Base‘𝑇))
284283feq3d 6722 . . . . . . . . . 10 (𝜑 → ((1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶𝑉 ↔ (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇)))
285282, 284mpbird 257 . . . . . . . . 9 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶𝑉)
286285fovcdmda 7605 . . . . . . . 8 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((1st𝑍)𝑤) ∈ 𝑉)
287 1st2ndbr 8068 . . . . . . . . . . . 12 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
288279, 23, 287sylancr 587 . . . . . . . . . . 11 (𝜑 → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
2898, 278, 288funcf1 17912 . . . . . . . . . 10 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
290283feq3d 6722 . . . . . . . . . 10 (𝜑 → ((1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶𝑉 ↔ (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇)))
291289, 290mpbird 257 . . . . . . . . 9 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶𝑉)
292291fovcdmda 7605 . . . . . . . 8 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((1st𝐸)𝑤) ∈ 𝑉)
29313, 29, 286, 292, 25setcinv 18136 . . . . . . 7 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → ((𝑀𝑤)(((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤))(𝑁𝑤) ↔ ((𝑀𝑤):((1st𝑍)𝑤)–1-1-onto→((1st𝐸)𝑤) ∧ (𝑁𝑤) = (𝑀𝑤))))
294277, 293mpbird 257 . . . . . 6 ((𝜑 ∧ ( ∈ (𝑂 Func 𝑆) ∧ 𝑤𝐵)) → (𝑀𝑤)(((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤))(𝑁𝑤))
295294ralrimivva 3201 . . . . 5 (𝜑 → ∀ ∈ (𝑂 Func 𝑆)∀𝑤𝐵 (𝑀𝑤)(((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤))(𝑁𝑤))
296 fveq2 6905 . . . . . . . 8 (𝑧 = ⟨, 𝑤⟩ → (𝑀𝑧) = (𝑀‘⟨, 𝑤⟩))
297 df-ov 7435 . . . . . . . 8 (𝑀𝑤) = (𝑀‘⟨, 𝑤⟩)
298296, 297eqtr4di 2794 . . . . . . 7 (𝑧 = ⟨, 𝑤⟩ → (𝑀𝑧) = (𝑀𝑤))
299 fveq2 6905 . . . . . . . . 9 (𝑧 = ⟨, 𝑤⟩ → ((1st𝑍)‘𝑧) = ((1st𝑍)‘⟨, 𝑤⟩))
300 df-ov 7435 . . . . . . . . 9 ((1st𝑍)𝑤) = ((1st𝑍)‘⟨, 𝑤⟩)
301299, 300eqtr4di 2794 . . . . . . . 8 (𝑧 = ⟨, 𝑤⟩ → ((1st𝑍)‘𝑧) = ((1st𝑍)𝑤))
302 fveq2 6905 . . . . . . . . 9 (𝑧 = ⟨, 𝑤⟩ → ((1st𝐸)‘𝑧) = ((1st𝐸)‘⟨, 𝑤⟩))
303 df-ov 7435 . . . . . . . . 9 ((1st𝐸)𝑤) = ((1st𝐸)‘⟨, 𝑤⟩)
304302, 303eqtr4di 2794 . . . . . . . 8 (𝑧 = ⟨, 𝑤⟩ → ((1st𝐸)‘𝑧) = ((1st𝐸)𝑤))
305301, 304oveq12d 7450 . . . . . . 7 (𝑧 = ⟨, 𝑤⟩ → (((1st𝑍)‘𝑧)(Inv‘𝑇)((1st𝐸)‘𝑧)) = (((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤)))
306 fveq2 6905 . . . . . . . 8 (𝑧 = ⟨, 𝑤⟩ → (𝑁𝑧) = (𝑁‘⟨, 𝑤⟩))
307 df-ov 7435 . . . . . . . 8 (𝑁𝑤) = (𝑁‘⟨, 𝑤⟩)
308306, 307eqtr4di 2794 . . . . . . 7 (𝑧 = ⟨, 𝑤⟩ → (𝑁𝑧) = (𝑁𝑤))
309298, 305, 308breq123d 5156 . . . . . 6 (𝑧 = ⟨, 𝑤⟩ → ((𝑀𝑧)(((1st𝑍)‘𝑧)(Inv‘𝑇)((1st𝐸)‘𝑧))(𝑁𝑧) ↔ (𝑀𝑤)(((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤))(𝑁𝑤)))
310309ralxp 5851 . . . . 5 (∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧)(((1st𝑍)‘𝑧)(Inv‘𝑇)((1st𝐸)‘𝑧))(𝑁𝑧) ↔ ∀ ∈ (𝑂 Func 𝑆)∀𝑤𝐵 (𝑀𝑤)(((1st𝑍)𝑤)(Inv‘𝑇)((1st𝐸)𝑤))(𝑁𝑤))
311295, 310sylibr 234 . . . 4 (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧)(((1st𝑍)‘𝑧)(Inv‘𝑇)((1st𝐸)‘𝑧))(𝑁𝑧))
312311r19.21bi 3250 . . 3 ((𝜑𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)) → (𝑀𝑧)(((1st𝑍)‘𝑧)(Inv‘𝑇)((1st𝐸)‘𝑧))(𝑁𝑧))
3131, 8, 9, 22, 23, 24, 25, 27, 312invfuc 18023 . 2 (𝜑𝑀(𝑍𝐼𝐸)(𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ↦ (𝑁𝑧)))
314 fvex 6918 . . . . 5 ((1st𝑓)‘𝑥) ∈ V
315314mptex 7244 . . . 4 (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))) ∈ V
31642, 315fnmpoi 8096 . . 3 𝑁 Fn ((𝑂 Func 𝑆) × 𝐵)
317 dffn5 6966 . . 3 (𝑁 Fn ((𝑂 Func 𝑆) × 𝐵) ↔ 𝑁 = (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ↦ (𝑁𝑧)))
318316, 317mpbi 230 . 2 𝑁 = (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ↦ (𝑁𝑧))
319313, 318breqtrrdi 5184 1 (𝜑𝑀(𝑍𝐼𝐸)𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  cun 3948  wss 3950  cop 4631   class class class wbr 5142  cmpt 5224   I cid 5576   × cxp 5682  ccnv 5683  ran crn 5685  cres 5686  ccom 5688  Rel wrel 5689   Fn wfn 6555  wf 6556  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  cmpo 7434  1st c1st 8013  2nd c2nd 8014  tpos ctpos 8251  Basecbs 17248  Hom chom 17309  compcco 17310  Catccat 17708  Idccid 17709  Homf chomf 17710  oppCatcoppc 17755  Invcinv 17790   Func cfunc 17900  func ccofu 17902   Nat cnat 17990   FuncCat cfuc 17991  SetCatcsetc 18121   ×c cxpc 18214   1stF c1stf 18215   2ndF c2ndf 18216   ⟨,⟩F cprf 18217   evalF cevlf 18255  HomFchof 18294  Yoncyon 18295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-hom 17322  df-cco 17323  df-cat 17712  df-cid 17713  df-homf 17714  df-comf 17715  df-oppc 17756  df-sect 17792  df-inv 17793  df-ssc 17855  df-resc 17856  df-subc 17857  df-func 17904  df-cofu 17906  df-nat 17992  df-fuc 17993  df-setc 18122  df-xpc 18218  df-1stf 18219  df-2ndf 18220  df-prf 18221  df-evlf 18259  df-curf 18260  df-hof 18296  df-yon 18297
This theorem is referenced by:  yonffthlem  18328  yoneda  18329
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