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Theorem cofuval 16365
 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 16343 . . 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 790 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
43fveq2d 6107 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑔) = (1st𝐺))
5 simprr 792 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
65fveq2d 6107 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑓) = (1st𝐹))
74, 6coeq12d 5208 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑔) ∘ (1st𝑓)) = ((1st𝐺) ∘ (1st𝐹)))
85fveq2d 6107 . . . . . . . 8 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
98dmeqd 5248 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝑓) = dom (2nd𝐹))
10 cofuval.b . . . . . . . . . 10 𝐵 = (Base‘𝐶)
11 relfunc 16345 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
12 cofuval.f . . . . . . . . . . 11 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
13 1st2ndbr 7108 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1411, 12, 13sylancr 694 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1510, 14funcfn2 16352 . . . . . . . . 9 (𝜑 → (2nd𝐹) Fn (𝐵 × 𝐵))
16 fndm 5904 . . . . . . . . 9 ((2nd𝐹) Fn (𝐵 × 𝐵) → dom (2nd𝐹) = (𝐵 × 𝐵))
1715, 16syl 17 . . . . . . . 8 (𝜑 → dom (2nd𝐹) = (𝐵 × 𝐵))
1817adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝐹) = (𝐵 × 𝐵))
199, 18eqtrd 2644 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom (2nd𝑓) = (𝐵 × 𝐵))
2019dmeqd 5248 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom dom (2nd𝑓) = dom (𝐵 × 𝐵))
21 dmxpid 5266 . . . . 5 dom (𝐵 × 𝐵) = 𝐵
2220, 21syl6eq 2660 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → dom dom (2nd𝑓) = 𝐵)
233fveq2d 6107 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
246fveq1d 6105 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑥))
256fveq1d 6105 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑓)‘𝑦) = ((1st𝐹)‘𝑦))
2623, 24, 25oveq123d 6570 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) = (((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)))
278oveqd 6566 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑥(2nd𝑓)𝑦) = (𝑥(2nd𝐹)𝑦))
2826, 27coeq12d 5208 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)) = ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))
2922, 22, 28mpt2eq123dv 6615 . . 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 4348 . 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 3185 . . 3 (𝐺 ∈ (𝐷 Func 𝐸) → 𝐺 ∈ V)
3331, 32syl 17 . 2 (𝜑𝐺 ∈ V)
34 elex 3185 . . 3 (𝐹 ∈ (𝐶 Func 𝐷) → 𝐹 ∈ V)
3512, 34syl 17 . 2 (𝜑𝐹 ∈ V)
36 opex 4859 . . 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 6686 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 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038   ∘ ccom 5042  Rel wrel 5043   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Basecbs 15695   Func cfunc 16337   ∘func ccofu 16339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-func 16341  df-cofu 16343 This theorem is referenced by:  cofu1st  16366  cofu2nd  16368  cofuval2  16370  cofucl  16371  cofuass  16372  cofulid  16373  cofurid  16374  prf1st  16667  prf2nd  16668
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