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Theorem cofuval 16981
 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 16959 . . 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 767 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
43fveq2d 6542 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑔) = (1st𝐺))
5 simprr 769 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
65fveq2d 6542 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑓) = (1st𝐹))
74, 6coeq12d 5621 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑔) ∘ (1st𝑓)) = ((1st𝐺) ∘ (1st𝐹)))
85fveq2d 6542 . . . . . . . 8 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
98dmeqd 5660 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝑓) = dom (2nd𝐹))
10 cofuval.b . . . . . . . . . 10 𝐵 = (Base‘𝐶)
11 relfunc 16961 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
12 cofuval.f . . . . . . . . . . 11 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
13 1st2ndbr 7597 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1411, 12, 13sylancr 587 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1510, 14funcfn2 16968 . . . . . . . . 9 (𝜑 → (2nd𝐹) Fn (𝐵 × 𝐵))
16 fndm 6325 . . . . . . . . 9 ((2nd𝐹) Fn (𝐵 × 𝐵) → dom (2nd𝐹) = (𝐵 × 𝐵))
1715, 16syl 17 . . . . . . . 8 (𝜑 → dom (2nd𝐹) = (𝐵 × 𝐵))
1817adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝐹) = (𝐵 × 𝐵))
199, 18eqtrd 2831 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝑓) = (𝐵 × 𝐵))
2019dmeqd 5660 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom dom (2nd𝑓) = dom (𝐵 × 𝐵))
21 dmxpid 5682 . . . . 5 dom (𝐵 × 𝐵) = 𝐵
2220, 21syl6eq 2847 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom dom (2nd𝑓) = 𝐵)
233fveq2d 6542 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
246fveq1d 6540 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑥))
256fveq1d 6540 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑓)‘𝑦) = ((1st𝐹)‘𝑦))
2623, 24, 25oveq123d 7037 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) = (((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)))
278oveqd 7033 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑥(2nd𝑓)𝑦) = (𝑥(2nd𝐹)𝑦))
2826, 27coeq12d 5621 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)) = ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))
2922, 22, 28mpoeq123dv 7087 . . 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 4718 . 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))
3231elexd 3457 . 2 (𝜑𝐺 ∈ V)
3312elexd 3457 . 2 (𝜑𝐹 ∈ V)
34 opex 5248 . . 3 ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩ ∈ V
3534a1i 11 . 2 (𝜑 → ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩ ∈ V)
362, 30, 32, 33, 35ovmpod 7158 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 396   = wceq 1522   ∈ wcel 2081  Vcvv 3437  ⟨cop 4478   class class class wbr 4962   × cxp 5441  dom cdm 5443   ∘ ccom 5447  Rel wrel 5448   Fn wfn 6220  ‘cfv 6225  (class class class)co 7016   ∈ cmpo 7018  1st c1st 7543  2nd c2nd 7544  Basecbs 16312   Func cfunc 16953   ∘func ccofu 16955 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-map 8258  df-ixp 8311  df-func 16957  df-cofu 16959 This theorem is referenced by:  cofu1st  16982  cofu2nd  16984  cofuval2  16986  cofucl  16987  cofuass  16988  cofulid  16989  cofurid  16990  prf1st  17283  prf2nd  17284
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