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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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