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Theorem cofu2nd 16746
 Description: Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofuval.b 𝐵 = (Base‘𝐶)
cofuval.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
cofuval.g (𝜑𝐺 ∈ (𝐷 Func 𝐸))
cofu2nd.x (𝜑𝑋𝐵)
cofu2nd.y (𝜑𝑌𝐵)
Assertion
Ref Expression
cofu2nd (𝜑 → (𝑋(2nd ‘(𝐺func 𝐹))𝑌) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))

Proof of Theorem cofu2nd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cofuval.b . . . . 5 𝐵 = (Base‘𝐶)
2 cofuval.f . . . . 5 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
3 cofuval.g . . . . 5 (𝜑𝐺 ∈ (𝐷 Func 𝐸))
41, 2, 3cofuval 16743 . . . 4 (𝜑 → (𝐺func 𝐹) = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
54fveq2d 6356 . . 3 (𝜑 → (2nd ‘(𝐺func 𝐹)) = (2nd ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩))
6 fvex 6362 . . . . 5 (1st𝐺) ∈ V
7 fvex 6362 . . . . 5 (1st𝐹) ∈ V
86, 7coex 7283 . . . 4 ((1st𝐺) ∘ (1st𝐹)) ∈ V
9 fvex 6362 . . . . . 6 (Base‘𝐶) ∈ V
101, 9eqeltri 2835 . . . . 5 𝐵 ∈ V
1110, 10mpt2ex 7415 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))) ∈ V
128, 11op2nd 7342 . . 3 (2nd ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩) = (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))
135, 12syl6eq 2810 . 2 (𝜑 → (2nd ‘(𝐺func 𝐹)) = (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))))
14 simprl 811 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
1514fveq2d 6356 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑋))
16 simprr 813 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
1716fveq2d 6356 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((1st𝐹)‘𝑦) = ((1st𝐹)‘𝑌))
1815, 17oveq12d 6831 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) = (((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)))
1914, 16oveq12d 6831 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥(2nd𝐹)𝑦) = (𝑋(2nd𝐹)𝑌))
2018, 19coeq12d 5442 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))
21 cofu2nd.x . 2 (𝜑𝑋𝐵)
22 cofu2nd.y . 2 (𝜑𝑌𝐵)
23 ovex 6841 . . . 4 (((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∈ V
24 ovex 6841 . . . 4 (𝑋(2nd𝐹)𝑌) ∈ V
2523, 24coex 7283 . . 3 ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)) ∈ V
2625a1i 11 . 2 (𝜑 → ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)) ∈ V)
2713, 20, 21, 22, 26ovmpt2d 6953 1 (𝜑 → (𝑋(2nd ‘(𝐺func 𝐹))𝑌) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ⟨cop 4327   ∘ ccom 5270  ‘cfv 6049  (class class class)co 6813   ↦ cmpt2 6815  1st c1st 7331  2nd c2nd 7332  Basecbs 16059   Func cfunc 16715   ∘func ccofu 16717 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-ixp 8075  df-func 16719  df-cofu 16721 This theorem is referenced by:  cofu2  16747  cofucl  16749  cofuass  16750  cofull  16795  cofth  16796  catciso  16958
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