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Theorem ofoprabco 29304
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)
32ffvelrnda 6315 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) ∈ 𝐵)
4 ofoprabco.3 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
54ffvelrnda 6315 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐺𝑎) ∈ 𝐶)
6 opelxpi 5108 . . . . . . 7 (((𝐹𝑎) ∈ 𝐵 ∧ (𝐺𝑎) ∈ 𝐶) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (𝐵 × 𝐶))
73, 5, 6syl2anc 692 . . . . . 6 ((𝜑𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (𝐵 × 𝐶))
81, 7fvmpt2d 6250 . . . . 5 ((𝜑𝑎𝐴) → (𝑀𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
98fveq2d 6152 . . . 4 ((𝜑𝑎𝐴) → (𝑁‘(𝑀𝑎)) = (𝑁‘⟨(𝐹𝑎), (𝐺𝑎)⟩))
10 df-ov 6607 . . . . 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 481 . . . . 5 ((𝜑𝑎𝐴) → 𝑁 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)))
14 simprl 793 . . . . . 6 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → 𝑥 = (𝐹𝑎))
15 simprr 795 . . . . . 6 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → 𝑦 = (𝐺𝑎))
1614, 15oveq12d 6622 . . . . 5 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → (𝑥𝑅𝑦) = ((𝐹𝑎)𝑅(𝐺𝑎)))
17 ovex 6632 . . . . . 6 ((𝐹𝑎)𝑅(𝐺𝑎)) ∈ V
1817a1i 11 . . . . 5 ((𝜑𝑎𝐴) → ((𝐹𝑎)𝑅(𝐺𝑎)) ∈ V)
1913, 16, 3, 5, 18ovmpt2d 6741 . . . 4 ((𝜑𝑎𝐴) → ((𝐹𝑎)𝑁(𝐺𝑎)) = ((𝐹𝑎)𝑅(𝐺𝑎)))
209, 11, 193eqtr2d 2661 . . 3 ((𝜑𝑎𝐴) → (𝑁‘(𝑀𝑎)) = ((𝐹𝑎)𝑅(𝐺𝑎)))
2120mpteq2dva 4704 . 2 (𝜑 → (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎)𝑅(𝐺𝑎))))
22 ovex 6632 . . . . . 6 (𝑥𝑅𝑦) ∈ V
2322rgen2w 2920 . . . . 5 𝑥𝐵𝑦𝐶 (𝑥𝑅𝑦) ∈ V
24 eqid 2621 . . . . . 6 (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)) = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦))
2524fmpt2 7182 . . . . 5 (∀𝑥𝐵𝑦𝐶 (𝑥𝑅𝑦) ∈ V ↔ (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V)
2623, 25mpbi 220 . . . 4 (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V
2712feq1d 5987 . . . 4 (𝜑 → (𝑁:(𝐵 × 𝐶)⟶V ↔ (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V))
2826, 27mpbiri 248 . . 3 (𝜑𝑁:(𝐵 × 𝐶)⟶V)
291, 7fmpt3d 6341 . . 3 (𝜑𝑀:𝐴⟶(𝐵 × 𝐶))
30 ofoprabco.1 . . . 4 𝑎𝑀
3130fcomptf 29297 . . 3 ((𝑁:(𝐵 × 𝐶)⟶V ∧ 𝑀:𝐴⟶(𝐵 × 𝐶)) → (𝑁𝑀) = (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))))
3228, 29, 31syl2anc 692 . 2 (𝜑 → (𝑁𝑀) = (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))))
33 ofoprabco.4 . . 3 (𝜑𝐴𝑉)
342feqmptd 6206 . . 3 (𝜑𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
354feqmptd 6206 . . 3 (𝜑𝐺 = (𝑎𝐴 ↦ (𝐺𝑎)))
3633, 3, 5, 34, 35offval2 6867 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎)𝑅(𝐺𝑎))))
3721, 32, 363eqtr4rd 2666 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑁𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wnfc 2748  wral 2907  Vcvv 3186  cop 4154  cmpt 4673   × cxp 5072  ccom 5078  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑓 cof 6848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-1st 7113  df-2nd 7114
This theorem is referenced by:  ofpreima  29305  rrvadd  30292
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