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Theorem ofoprabco 31580
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 7035 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) ∈ 𝐵)
4 ofoprabco.3 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
54ffvelcdmda 7035 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐺𝑎) ∈ 𝐶)
6 opelxpi 5670 . . . . . . 7 (((𝐹𝑎) ∈ 𝐵 ∧ (𝐺𝑎) ∈ 𝐶) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (𝐵 × 𝐶))
73, 5, 6syl2anc 584 . . . . . 6 ((𝜑𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (𝐵 × 𝐶))
81, 7fvmpt2d 6961 . . . . 5 ((𝜑𝑎𝐴) → (𝑀𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
98fveq2d 6846 . . . 4 ((𝜑𝑎𝐴) → (𝑁‘(𝑀𝑎)) = (𝑁‘⟨(𝐹𝑎), (𝐺𝑎)⟩))
10 df-ov 7360 . . . . 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 769 . . . . . 6 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → 𝑥 = (𝐹𝑎))
15 simprr 771 . . . . . 6 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → 𝑦 = (𝐺𝑎))
1614, 15oveq12d 7375 . . . . 5 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → (𝑥𝑅𝑦) = ((𝐹𝑎)𝑅(𝐺𝑎)))
17 ovexd 7392 . . . . 5 ((𝜑𝑎𝐴) → ((𝐹𝑎)𝑅(𝐺𝑎)) ∈ V)
1813, 16, 3, 5, 17ovmpod 7507 . . . 4 ((𝜑𝑎𝐴) → ((𝐹𝑎)𝑁(𝐺𝑎)) = ((𝐹𝑎)𝑅(𝐺𝑎)))
199, 11, 183eqtr2d 2782 . . 3 ((𝜑𝑎𝐴) → (𝑁‘(𝑀𝑎)) = ((𝐹𝑎)𝑅(𝐺𝑎)))
2019mpteq2dva 5205 . 2 (𝜑 → (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎)𝑅(𝐺𝑎))))
21 ovex 7390 . . . . . 6 (𝑥𝑅𝑦) ∈ V
2221rgen2w 3069 . . . . 5 𝑥𝐵𝑦𝐶 (𝑥𝑅𝑦) ∈ V
23 eqid 2736 . . . . . 6 (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)) = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦))
2423fmpo 8000 . . . . 5 (∀𝑥𝐵𝑦𝐶 (𝑥𝑅𝑦) ∈ V ↔ (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V)
2522, 24mpbi 229 . . . 4 (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V
2612feq1d 6653 . . . 4 (𝜑 → (𝑁:(𝐵 × 𝐶)⟶V ↔ (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V))
2725, 26mpbiri 257 . . 3 (𝜑𝑁:(𝐵 × 𝐶)⟶V)
281, 7fmpt3d 7064 . . 3 (𝜑𝑀:𝐴⟶(𝐵 × 𝐶))
29 ofoprabco.1 . . . 4 𝑎𝑀
3029fcomptf 31574 . . 3 ((𝑁:(𝐵 × 𝐶)⟶V ∧ 𝑀:𝐴⟶(𝐵 × 𝐶)) → (𝑁𝑀) = (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))))
3127, 28, 30syl2anc 584 . 2 (𝜑 → (𝑁𝑀) = (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))))
32 ofoprabco.4 . . 3 (𝜑𝐴𝑉)
332feqmptd 6910 . . 3 (𝜑𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
344feqmptd 6910 . . 3 (𝜑𝐺 = (𝑎𝐴 ↦ (𝐺𝑎)))
3532, 3, 5, 33, 34offval2 7637 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎)𝑅(𝐺𝑎))))
3620, 31, 353eqtr4rd 2787 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝑁𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wnfc 2887  wral 3064  Vcvv 3445  cop 4592  cmpt 5188   × cxp 5631  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  f cof 7615
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-1st 7921  df-2nd 7922
This theorem is referenced by:  ofpreima  31581  rrvadd  33052
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