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Theorem cofu2nd 16314
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 16311 . . . 4 (𝜑 → (𝐺func 𝐹) = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
54fveq2d 6092 . . 3 (𝜑 → (2nd ‘(𝐺func 𝐹)) = (2nd ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩))
6 fvex 6098 . . . . 5 (1st𝐺) ∈ V
7 fvex 6098 . . . . 5 (1st𝐹) ∈ V
86, 7coex 6988 . . . 4 ((1st𝐺) ∘ (1st𝐹)) ∈ V
9 fvex 6098 . . . . . 6 (Base‘𝐶) ∈ V
101, 9eqeltri 2683 . . . . 5 𝐵 ∈ V
1110, 10mpt2ex 7113 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))) ∈ V
128, 11op2nd 7045 . . 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 2659 . 2 (𝜑 → (2nd ‘(𝐺func 𝐹)) = (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))))
14 simprl 789 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
1514fveq2d 6092 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑋))
16 simprr 791 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
1716fveq2d 6092 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((1st𝐹)‘𝑦) = ((1st𝐹)‘𝑌))
1815, 17oveq12d 6545 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) = (((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)))
1914, 16oveq12d 6545 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥(2nd𝐹)𝑦) = (𝑋(2nd𝐹)𝑌))
2018, 19coeq12d 5196 . 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 6555 . . . 4 (((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∈ V
24 ovex 6555 . . . 4 (𝑋(2nd𝐹)𝑌) ∈ V
2523, 24coex 6988 . . 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 6664 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 382   = wceq 1474  wcel 1976  Vcvv 3172  cop 4130  ccom 5032  cfv 5790  (class class class)co 6527  cmpt2 6529  1st c1st 7034  2nd c2nd 7035  Basecbs 15641   Func cfunc 16283  func ccofu 16285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-map 7723  df-ixp 7772  df-func 16287  df-cofu 16289
This theorem is referenced by:  cofu2  16315  cofucl  16317  cofuass  16318  cofull  16363  cofth  16364  catciso  16526
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