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Theorem cofu1 16525
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 16524 . . 3 (𝜑 → (1st ‘(𝐺func 𝐹)) = ((1st𝐺) ∘ (1st𝐹)))
54fveq1d 6180 . 2 (𝜑 → ((1st ‘(𝐺func 𝐹))‘𝑋) = (((1st𝐺) ∘ (1st𝐹))‘𝑋))
6 eqid 2620 . . . 4 (Base‘𝐷) = (Base‘𝐷)
7 relfunc 16503 . . . . 5 Rel (𝐶 Func 𝐷)
8 1st2ndbr 7202 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
97, 2, 8sylancr 694 . . . 4 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
101, 6, 9funcf1 16507 . . 3 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
11 cofu2nd.x . . 3 (𝜑𝑋𝐵)
12 fvco3 6262 . . 3 (((1st𝐹):𝐵⟶(Base‘𝐷) ∧ 𝑋𝐵) → (((1st𝐺) ∘ (1st𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
1310, 11, 12syl2anc 692 . 2 (𝜑 → (((1st𝐺) ∘ (1st𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
145, 13eqtrd 2654 1 (𝜑 → ((1st ‘(𝐺func 𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988   class class class wbr 4644  ccom 5108  Rel wrel 5109  wf 5872  cfv 5876  (class class class)co 6635  1st c1st 7151  2nd c2nd 7152  Basecbs 15838   Func cfunc 16495  func ccofu 16497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844  df-ixp 7894  df-func 16499  df-cofu 16501
This theorem is referenced by:  cofucl  16529  cofuass  16530  cofull  16575  cofth  16576  catciso  16738  1st2ndprf  16827  uncf1  16857  uncf2  16858  yonedalem21  16894  yonedalem22  16899
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