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Theorem ofoprabco 31286
Description: Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)
Hypotheses
Ref Expression
ofoprabco.1 𝑎𝑀
ofoprabco.2 (𝜑𝐹:𝐴𝐵)
ofoprabco.3 (𝜑𝐺:𝐴𝐶)
ofoprabco.4 (𝜑𝐴𝑉)
ofoprabco.5 (𝜑𝑀 = (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩))
ofoprabco.6 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)))
Assertion
Ref Expression
ofoprabco (𝜑 → (𝐹f 𝑅𝐺) = (𝑁𝑀))
Distinct variable groups:   𝑥,𝑎,𝑦,𝐴   𝐵,𝑎,𝑥,𝑦   𝐶,𝑎,𝑥,𝑦   𝐹,𝑎,𝑥,𝑦   𝐺,𝑎,𝑥,𝑦   𝑁,𝑎   𝑅,𝑎,𝑥,𝑦   𝜑,𝑎,𝑥,𝑦
Allowed substitution hints:   𝑀(𝑥,𝑦,𝑎)   𝑁(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑎)

Proof of Theorem ofoprabco
StepHypRef Expression
1 ofoprabco.5 . . . . . 6 (𝜑𝑀 = (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩))
2 ofoprabco.2 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
32ffvelcdmda 7022 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) ∈ 𝐵)
4 ofoprabco.3 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
54ffvelcdmda 7022 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐺𝑎) ∈ 𝐶)
6 opelxpi 5662 . . . . . . 7 (((𝐹𝑎) ∈ 𝐵 ∧ (𝐺𝑎) ∈ 𝐶) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (𝐵 × 𝐶))
73, 5, 6syl2anc 585 . . . . . 6 ((𝜑𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (𝐵 × 𝐶))
81, 7fvmpt2d 6949 . . . . 5 ((𝜑𝑎𝐴) → (𝑀𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
98fveq2d 6834 . . . 4 ((𝜑𝑎𝐴) → (𝑁‘(𝑀𝑎)) = (𝑁‘⟨(𝐹𝑎), (𝐺𝑎)⟩))
10 df-ov 7345 . . . . 5 ((𝐹𝑎)𝑁(𝐺𝑎)) = (𝑁‘⟨(𝐹𝑎), (𝐺𝑎)⟩)
1110a1i 11 . . . 4 ((𝜑𝑎𝐴) → ((𝐹𝑎)𝑁(𝐺𝑎)) = (𝑁‘⟨(𝐹𝑎), (𝐺𝑎)⟩))
12 ofoprabco.6 . . . . . 6 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)))
1312adantr 482 . . . . 5 ((𝜑𝑎𝐴) → 𝑁 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)))
14 simprl 769 . . . . . 6 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → 𝑥 = (𝐹𝑎))
15 simprr 771 . . . . . 6 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → 𝑦 = (𝐺𝑎))
1614, 15oveq12d 7360 . . . . 5 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → (𝑥𝑅𝑦) = ((𝐹𝑎)𝑅(𝐺𝑎)))
17 ovexd 7377 . . . . 5 ((𝜑𝑎𝐴) → ((𝐹𝑎)𝑅(𝐺𝑎)) ∈ V)
1813, 16, 3, 5, 17ovmpod 7492 . . . 4 ((𝜑𝑎𝐴) → ((𝐹𝑎)𝑁(𝐺𝑎)) = ((𝐹𝑎)𝑅(𝐺𝑎)))
199, 11, 183eqtr2d 2783 . . 3 ((𝜑𝑎𝐴) → (𝑁‘(𝑀𝑎)) = ((𝐹𝑎)𝑅(𝐺𝑎)))
2019mpteq2dva 5197 . 2 (𝜑 → (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎)𝑅(𝐺𝑎))))
21 ovex 7375 . . . . . 6 (𝑥𝑅𝑦) ∈ V
2221rgen2w 3067 . . . . 5 𝑥𝐵𝑦𝐶 (𝑥𝑅𝑦) ∈ V
23 eqid 2737 . . . . . 6 (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)) = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦))
2423fmpo 7981 . . . . 5 (∀𝑥𝐵𝑦𝐶 (𝑥𝑅𝑦) ∈ V ↔ (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V)
2522, 24mpbi 229 . . . 4 (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V
2612feq1d 6641 . . . 4 (𝜑 → (𝑁:(𝐵 × 𝐶)⟶V ↔ (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V))
2725, 26mpbiri 258 . . 3 (𝜑𝑁:(𝐵 × 𝐶)⟶V)
281, 7fmpt3d 7051 . . 3 (𝜑𝑀:𝐴⟶(𝐵 × 𝐶))
29 ofoprabco.1 . . . 4 𝑎𝑀
3029fcomptf 31280 . . 3 ((𝑁:(𝐵 × 𝐶)⟶V ∧ 𝑀:𝐴⟶(𝐵 × 𝐶)) → (𝑁𝑀) = (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))))
3127, 28, 30syl2anc 585 . 2 (𝜑 → (𝑁𝑀) = (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))))
32 ofoprabco.4 . . 3 (𝜑𝐴𝑉)
332feqmptd 6898 . . 3 (𝜑𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
344feqmptd 6898 . . 3 (𝜑𝐺 = (𝑎𝐴 ↦ (𝐺𝑎)))
3532, 3, 5, 33, 34offval2 7620 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎)𝑅(𝐺𝑎))))
3620, 31, 353eqtr4rd 2788 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝑁𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1541  wcel 2106  wnfc 2885  wral 3062  Vcvv 3442  cop 4584  cmpt 5180   × cxp 5623  ccom 5629  wf 6480  cfv 6484  (class class class)co 7342  cmpo 7344  f cof 7598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pr 5377  ax-un 7655
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-ov 7345  df-oprab 7346  df-mpo 7347  df-of 7600  df-1st 7904  df-2nd 7905
This theorem is referenced by:  ofpreima  31287  rrvadd  32717
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