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Theorem cofu1st 17144
 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 17143 . . 3 (𝜑 → (𝐺func 𝐹) = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
54fveq2d 6656 . 2 (𝜑 → (1st ‘(𝐺func 𝐹)) = (1st ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩))
6 fvex 6665 . . . 4 (1st𝐺) ∈ V
7 fvex 6665 . . . 4 (1st𝐹) ∈ V
86, 7coex 7621 . . 3 ((1st𝐺) ∘ (1st𝐹)) ∈ V
91fvexi 6666 . . . 4 𝐵 ∈ V
109, 9mpoex 7764 . . 3 (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))) ∈ V
118, 10op1st 7683 . 2 (1st ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩) = ((1st𝐺) ∘ (1st𝐹))
125, 11syl6eq 2873 1 (𝜑 → (1st ‘(𝐺func 𝐹)) = ((1st𝐺) ∘ (1st𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2114  ⟨cop 4545   ∘ ccom 5536  ‘cfv 6334  (class class class)co 7140   ∈ cmpo 7142  1st c1st 7673  2nd c2nd 7674  Basecbs 16474   Func cfunc 17115   ∘func ccofu 17117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-map 8395  df-ixp 8449  df-func 17119  df-cofu 17121 This theorem is referenced by:  cofu1  17145  cofucl  17149  cofuass  17150  catciso  17358
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