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Theorem cofu1 16929
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 16928 . . 3 (𝜑 → (1st ‘(𝐺func 𝐹)) = ((1st𝐺) ∘ (1st𝐹)))
54fveq1d 6448 . 2 (𝜑 → ((1st ‘(𝐺func 𝐹))‘𝑋) = (((1st𝐺) ∘ (1st𝐹))‘𝑋))
6 eqid 2777 . . . 4 (Base‘𝐷) = (Base‘𝐷)
7 relfunc 16907 . . . . 5 Rel (𝐶 Func 𝐷)
8 1st2ndbr 7496 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
97, 2, 8sylancr 581 . . . 4 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
101, 6, 9funcf1 16911 . . 3 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
11 cofu2nd.x . . 3 (𝜑𝑋𝐵)
12 fvco3 6535 . . 3 (((1st𝐹):𝐵⟶(Base‘𝐷) ∧ 𝑋𝐵) → (((1st𝐺) ∘ (1st𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
1310, 11, 12syl2anc 579 . 2 (𝜑 → (((1st𝐺) ∘ (1st𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
145, 13eqtrd 2813 1 (𝜑 → ((1st ‘(𝐺func 𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2106   class class class wbr 4886  ccom 5359  Rel wrel 5360  wf 6131  cfv 6135  (class class class)co 6922  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:  cofucl  16933  cofuass  16934  cofull  16979  cofth  16980  catciso  17142  1st2ndprf  17232  uncf1  17262  uncf2  17263  yonedalem21  17299  yonedalem22  17304
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