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Theorem cofu1 16896
Description: Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-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)
Assertion
Ref Expression
cofu1 (𝜑 → ((1st ‘(𝐺func 𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))

Proof of Theorem cofu1
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, 3cofu1st 16895 . . 3 (𝜑 → (1st ‘(𝐺func 𝐹)) = ((1st𝐺) ∘ (1st𝐹)))
54fveq1d 6435 . 2 (𝜑 → ((1st ‘(𝐺func 𝐹))‘𝑋) = (((1st𝐺) ∘ (1st𝐹))‘𝑋))
6 eqid 2825 . . . 4 (Base‘𝐷) = (Base‘𝐷)
7 relfunc 16874 . . . . 5 Rel (𝐶 Func 𝐷)
8 1st2ndbr 7479 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
97, 2, 8sylancr 583 . . . 4 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
101, 6, 9funcf1 16878 . . 3 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
11 cofu2nd.x . . 3 (𝜑𝑋𝐵)
12 fvco3 6522 . . 3 (((1st𝐹):𝐵⟶(Base‘𝐷) ∧ 𝑋𝐵) → (((1st𝐺) ∘ (1st𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
1310, 11, 12syl2anc 581 . 2 (𝜑 → (((1st𝐺) ∘ (1st𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
145, 13eqtrd 2861 1 (𝜑 → ((1st ‘(𝐺func 𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166   class class class wbr 4873  ccom 5346  Rel wrel 5347  wf 6119  cfv 6123  (class class class)co 6905  1st c1st 7426  2nd c2nd 7427  Basecbs 16222   Func cfunc 16866  func ccofu 16868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-map 8124  df-ixp 8176  df-func 16870  df-cofu 16872
This theorem is referenced by:  cofucl  16900  cofuass  16901  cofull  16946  cofth  16947  catciso  17109  1st2ndprf  17199  uncf1  17229  uncf2  17230  yonedalem21  17266  yonedalem22  17271
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