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Theorem cofu1st 16308
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 16307 . . 3 (𝜑 → (𝐺func 𝐹) = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
54fveq2d 6088 . 2 (𝜑 → (1st ‘(𝐺func 𝐹)) = (1st ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩))
6 fvex 6094 . . . 4 (1st𝐺) ∈ V
7 fvex 6094 . . . 4 (1st𝐹) ∈ V
86, 7coex 6984 . . 3 ((1st𝐺) ∘ (1st𝐹)) ∈ V
9 fvex 6094 . . . . 5 (Base‘𝐶) ∈ V
101, 9eqeltri 2679 . . . 4 𝐵 ∈ V
1110, 10mpt2ex 7109 . . 3 (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))) ∈ V
128, 11op1st 7040 . 2 (1st ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩) = ((1st𝐺) ∘ (1st𝐹))
135, 12syl6eq 2655 1 (𝜑 → (1st ‘(𝐺func 𝐹)) = ((1st𝐺) ∘ (1st𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  Vcvv 3168  cop 4126  ccom 5028  cfv 5786  (class class class)co 6523  cmpt2 6525  1st c1st 7030  2nd c2nd 7031  Basecbs 15637   Func cfunc 16279  func ccofu 16281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
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 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-map 7719  df-ixp 7768  df-func 16283  df-cofu 16285
This theorem is referenced by:  cofu1  16309  cofucl  16313  cofuass  16314  catciso  16522
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