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Theorem cofu2nd 16368
 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 16365 . . . 4 (𝜑 → (𝐺func 𝐹) = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
54fveq2d 6107 . . 3 (𝜑 → (2nd ‘(𝐺func 𝐹)) = (2nd ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩))
6 fvex 6113 . . . . 5 (1st𝐺) ∈ V
7 fvex 6113 . . . . 5 (1st𝐹) ∈ V
86, 7coex 7011 . . . 4 ((1st𝐺) ∘ (1st𝐹)) ∈ V
9 fvex 6113 . . . . . 6 (Base‘𝐶) ∈ V
101, 9eqeltri 2684 . . . . 5 𝐵 ∈ V
1110, 10mpt2ex 7136 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))) ∈ V
128, 11op2nd 7068 . . 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 2660 . 2 (𝜑 → (2nd ‘(𝐺func 𝐹)) = (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))))
14 simprl 790 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
1514fveq2d 6107 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑋))
16 simprr 792 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
1716fveq2d 6107 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((1st𝐹)‘𝑦) = ((1st𝐹)‘𝑌))
1815, 17oveq12d 6567 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) = (((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)))
1914, 16oveq12d 6567 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥(2nd𝐹)𝑦) = (𝑋(2nd𝐹)𝑌))
2018, 19coeq12d 5208 . 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 6577 . . . 4 (((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∈ V
24 ovex 6577 . . . 4 (𝑋(2nd𝐹)𝑌) ∈ V
2523, 24coex 7011 . . 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 6686 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 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Basecbs 15695   Func cfunc 16337   ∘func ccofu 16339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-func 16341  df-cofu 16343 This theorem is referenced by:  cofu2  16369  cofucl  16371  cofuass  16372  cofull  16417  cofth  16418  catciso  16580
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