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Theorem cofu1st 16928
 Description: Value of the object 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))
Assertion
Ref Expression
cofu1st (𝜑 → (1st ‘(𝐺func 𝐹)) = ((1st𝐺) ∘ (1st𝐹)))

Proof of Theorem cofu1st
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cofuval.b . . . 4 𝐵 = (Base‘𝐶)
2 cofuval.f . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
3 cofuval.g . . . 4 (𝜑𝐺 ∈ (𝐷 Func 𝐸))
41, 2, 3cofuval 16927 . . 3 (𝜑 → (𝐺func 𝐹) = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
54fveq2d 6450 . 2 (𝜑 → (1st ‘(𝐺func 𝐹)) = (1st ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩))
6 fvex 6459 . . . 4 (1st𝐺) ∈ V
7 fvex 6459 . . . 4 (1st𝐹) ∈ V
86, 7coex 7397 . . 3 ((1st𝐺) ∘ (1st𝐹)) ∈ V
91fvexi 6460 . . . 4 𝐵 ∈ V
109, 9mpt2ex 7527 . . 3 (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))) ∈ V
118, 10op1st 7453 . 2 (1st ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩) = ((1st𝐺) ∘ (1st𝐹))
125, 11syl6eq 2829 1 (𝜑 → (1st ‘(𝐺func 𝐹)) = ((1st𝐺) ∘ (1st𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2106  ⟨cop 4403   ∘ ccom 5359  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  1st c1st 7443  2nd c2nd 7444  Basecbs 16255   Func cfunc 16899   ∘func ccofu 16901 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-map 8142  df-ixp 8195  df-func 16903  df-cofu 16905 This theorem is referenced by:  cofu1  16929  cofucl  16933  cofuass  16934  catciso  17142
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