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Theorem cofuval 16749
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 16727 . . 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 754 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
43fveq2d 6336 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑔) = (1st𝐺))
5 simprr 756 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
65fveq2d 6336 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑓) = (1st𝐹))
74, 6coeq12d 5425 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑔) ∘ (1st𝑓)) = ((1st𝐺) ∘ (1st𝐹)))
85fveq2d 6336 . . . . . . . 8 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
98dmeqd 5464 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝑓) = dom (2nd𝐹))
10 cofuval.b . . . . . . . . . 10 𝐵 = (Base‘𝐶)
11 relfunc 16729 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
12 cofuval.f . . . . . . . . . . 11 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
13 1st2ndbr 7366 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1411, 12, 13sylancr 575 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1510, 14funcfn2 16736 . . . . . . . . 9 (𝜑 → (2nd𝐹) Fn (𝐵 × 𝐵))
16 fndm 6130 . . . . . . . . 9 ((2nd𝐹) Fn (𝐵 × 𝐵) → dom (2nd𝐹) = (𝐵 × 𝐵))
1715, 16syl 17 . . . . . . . 8 (𝜑 → dom (2nd𝐹) = (𝐵 × 𝐵))
1817adantr 466 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝐹) = (𝐵 × 𝐵))
199, 18eqtrd 2805 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝑓) = (𝐵 × 𝐵))
2019dmeqd 5464 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom dom (2nd𝑓) = dom (𝐵 × 𝐵))
21 dmxpid 5483 . . . . 5 dom (𝐵 × 𝐵) = 𝐵
2220, 21syl6eq 2821 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom dom (2nd𝑓) = 𝐵)
233fveq2d 6336 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
246fveq1d 6334 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑥))
256fveq1d 6334 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑓)‘𝑦) = ((1st𝐹)‘𝑦))
2623, 24, 25oveq123d 6814 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) = (((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)))
278oveqd 6810 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑥(2nd𝑓)𝑦) = (𝑥(2nd𝐹)𝑦))
2826, 27coeq12d 5425 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)) = ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))
2922, 22, 28mpt2eq123dv 6864 . . 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))))
307, 29opeq12d 4547 . 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)))⟩)
31 cofuval.g . . 3 (𝜑𝐺 ∈ (𝐷 Func 𝐸))
32 elex 3364 . . 3 (𝐺 ∈ (𝐷 Func 𝐸) → 𝐺 ∈ V)
3331, 32syl 17 . 2 (𝜑𝐺 ∈ V)
34 elex 3364 . . 3 (𝐹 ∈ (𝐶 Func 𝐷) → 𝐹 ∈ V)
3512, 34syl 17 . 2 (𝜑𝐹 ∈ V)
36 opex 5060 . . 3 ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩ ∈ V
3736a1i 11 . 2 (𝜑 → ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩ ∈ V)
382, 30, 33, 35, 37ovmpt2d 6935 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 382   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322   class class class wbr 4786   × cxp 5247  dom cdm 5249  ccom 5253  Rel wrel 5254   Fn wfn 6026  cfv 6031  (class class class)co 6793  cmpt2 6795  1st c1st 7313  2nd c2nd 7314  Basecbs 16064   Func cfunc 16721  func ccofu 16723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011  df-ixp 8063  df-func 16725  df-cofu 16727
This theorem is referenced by:  cofu1st  16750  cofu2nd  16752  cofuval2  16754  cofucl  16755  cofuass  16756  cofulid  16757  cofurid  16758  prf1st  17052  prf2nd  17053
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