MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cofu1 Structured version   Visualization version   GIF version

Theorem cofu1 17839
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 17838 . . 3 (𝜑 → (1st ‘(𝐺func 𝐹)) = ((1st𝐺) ∘ (1st𝐹)))
54fveq1d 6884 . 2 (𝜑 → ((1st ‘(𝐺func 𝐹))‘𝑋) = (((1st𝐺) ∘ (1st𝐹))‘𝑋))
6 eqid 2724 . . . 4 (Base‘𝐷) = (Base‘𝐷)
7 relfunc 17817 . . . . 5 Rel (𝐶 Func 𝐷)
8 1st2ndbr 8022 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
97, 2, 8sylancr 586 . . . 4 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
101, 6, 9funcf1 17821 . . 3 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
11 cofu2nd.x . . 3 (𝜑𝑋𝐵)
12 fvco3 6981 . . 3 (((1st𝐹):𝐵⟶(Base‘𝐷) ∧ 𝑋𝐵) → (((1st𝐺) ∘ (1st𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
1310, 11, 12syl2anc 583 . 2 (𝜑 → (((1st𝐺) ∘ (1st𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
145, 13eqtrd 2764 1 (𝜑 → ((1st ‘(𝐺func 𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098   class class class wbr 5139  ccom 5671  Rel wrel 5672  wf 6530  cfv 6534  (class class class)co 7402  1st c1st 7967  2nd c2nd 7968  Basecbs 17149   Func cfunc 17809  func ccofu 17811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-map 8819  df-ixp 8889  df-func 17813  df-cofu 17815
This theorem is referenced by:  cofucl  17843  cofuass  17844  cofull  17892  cofth  17893  catciso  18069  1st2ndprf  18166  uncf1  18197  uncf2  18198  yonedalem21  18234  yonedalem22  18239
  Copyright terms: Public domain W3C validator