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Theorem cofuval 17832
Description: Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofuval.b 𝐵 = (Base‘𝐶)
cofuval.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
cofuval.g (𝜑𝐺 ∈ (𝐷 Func 𝐸))
Assertion
Ref Expression
cofuval (𝜑 → (𝐺func 𝐹) = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)

Proof of Theorem cofuval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cofu 17810 . . 3 func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st𝑔) ∘ (1st𝑓)), (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)))⟩)
21a1i 11 . 2 (𝜑 → ∘func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st𝑔) ∘ (1st𝑓)), (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)))⟩))
3 simprl 770 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
43fveq2d 6896 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑔) = (1st𝐺))
5 simprr 772 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
65fveq2d 6896 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑓) = (1st𝐹))
74, 6coeq12d 5865 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑔) ∘ (1st𝑓)) = ((1st𝐺) ∘ (1st𝐹)))
85fveq2d 6896 . . . . . . . 8 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
98dmeqd 5906 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝑓) = dom (2nd𝐹))
10 cofuval.b . . . . . . . . . 10 𝐵 = (Base‘𝐶)
11 relfunc 17812 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
12 cofuval.f . . . . . . . . . . 11 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
13 1st2ndbr 8028 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1411, 12, 13sylancr 588 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1510, 14funcfn2 17819 . . . . . . . . 9 (𝜑 → (2nd𝐹) Fn (𝐵 × 𝐵))
1615fndmd 6655 . . . . . . . 8 (𝜑 → dom (2nd𝐹) = (𝐵 × 𝐵))
1716adantr 482 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝐹) = (𝐵 × 𝐵))
189, 17eqtrd 2773 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝑓) = (𝐵 × 𝐵))
1918dmeqd 5906 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom dom (2nd𝑓) = dom (𝐵 × 𝐵))
20 dmxpid 5930 . . . . 5 dom (𝐵 × 𝐵) = 𝐵
2119, 20eqtrdi 2789 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom dom (2nd𝑓) = 𝐵)
223fveq2d 6896 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
236fveq1d 6894 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑥))
246fveq1d 6894 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑓)‘𝑦) = ((1st𝐹)‘𝑦))
2522, 23, 24oveq123d 7430 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) = (((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)))
268oveqd 7426 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑥(2nd𝑓)𝑦) = (𝑥(2nd𝐹)𝑦))
2725, 26coeq12d 5865 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)) = ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))
2821, 21, 27mpoeq123dv 7484 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))))
297, 28opeq12d 4882 . 2 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨((1st𝑔) ∘ (1st𝑓)), (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)))⟩ = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
30 cofuval.g . . 3 (𝜑𝐺 ∈ (𝐷 Func 𝐸))
3130elexd 3495 . 2 (𝜑𝐺 ∈ V)
3212elexd 3495 . 2 (𝜑𝐹 ∈ V)
33 opex 5465 . . 3 ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩ ∈ V
3433a1i 11 . 2 (𝜑 → ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩ ∈ V)
352, 29, 31, 32, 34ovmpod 7560 1 (𝜑 → (𝐺func 𝐹) = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cop 4635   class class class wbr 5149   × cxp 5675  dom cdm 5677  ccom 5681  Rel wrel 5682  cfv 6544  (class class class)co 7409  cmpo 7411  1st c1st 7973  2nd c2nd 7974  Basecbs 17144   Func cfunc 17804  func ccofu 17806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-ixp 8892  df-func 17808  df-cofu 17810
This theorem is referenced by:  cofu1st  17833  cofu2nd  17835  cofuval2  17837  cofucl  17838  cofuass  17839  cofulid  17840  cofurid  17841  prf1st  18156  prf2nd  18157
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