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Theorem ofoprabco 30337
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 6843 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) ∈ 𝐵)
4 ofoprabco.3 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
54ffvelrnda 6843 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐺𝑎) ∈ 𝐶)
6 opelxpi 5585 . . . . . . 7 (((𝐹𝑎) ∈ 𝐵 ∧ (𝐺𝑎) ∈ 𝐶) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (𝐵 × 𝐶))
73, 5, 6syl2anc 584 . . . . . 6 ((𝜑𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (𝐵 × 𝐶))
81, 7fvmpt2d 6773 . . . . 5 ((𝜑𝑎𝐴) → (𝑀𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
98fveq2d 6667 . . . 4 ((𝜑𝑎𝐴) → (𝑁‘(𝑀𝑎)) = (𝑁‘⟨(𝐹𝑎), (𝐺𝑎)⟩))
10 df-ov 7148 . . . . 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 767 . . . . . 6 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → 𝑥 = (𝐹𝑎))
15 simprr 769 . . . . . 6 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → 𝑦 = (𝐺𝑎))
1614, 15oveq12d 7163 . . . . 5 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → (𝑥𝑅𝑦) = ((𝐹𝑎)𝑅(𝐺𝑎)))
17 ovexd 7180 . . . . 5 ((𝜑𝑎𝐴) → ((𝐹𝑎)𝑅(𝐺𝑎)) ∈ V)
1813, 16, 3, 5, 17ovmpod 7291 . . . 4 ((𝜑𝑎𝐴) → ((𝐹𝑎)𝑁(𝐺𝑎)) = ((𝐹𝑎)𝑅(𝐺𝑎)))
199, 11, 183eqtr2d 2859 . . 3 ((𝜑𝑎𝐴) → (𝑁‘(𝑀𝑎)) = ((𝐹𝑎)𝑅(𝐺𝑎)))
2019mpteq2dva 5152 . 2 (𝜑 → (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎)𝑅(𝐺𝑎))))
21 ovex 7178 . . . . . 6 (𝑥𝑅𝑦) ∈ V
2221rgen2w 3148 . . . . 5 𝑥𝐵𝑦𝐶 (𝑥𝑅𝑦) ∈ V
23 eqid 2818 . . . . . 6 (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)) = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦))
2423fmpo 7755 . . . . 5 (∀𝑥𝐵𝑦𝐶 (𝑥𝑅𝑦) ∈ V ↔ (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V)
2522, 24mpbi 231 . . . 4 (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V
2612feq1d 6492 . . . 4 (𝜑 → (𝑁:(𝐵 × 𝐶)⟶V ↔ (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V))
2725, 26mpbiri 259 . . 3 (𝜑𝑁:(𝐵 × 𝐶)⟶V)
281, 7fmpt3d 6872 . . 3 (𝜑𝑀:𝐴⟶(𝐵 × 𝐶))
29 ofoprabco.1 . . . 4 𝑎𝑀
3029fcomptf 30331 . . 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 6726 . . 3 (𝜑𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
344feqmptd 6726 . . 3 (𝜑𝐺 = (𝑎𝐴 ↦ (𝐺𝑎)))
3532, 3, 5, 33, 34offval2 7415 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎)𝑅(𝐺𝑎))))
3620, 31, 353eqtr4rd 2864 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝑁𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wnfc 2958  wral 3135  Vcvv 3492  cop 4563  cmpt 5137   × cxp 5546  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  f cof 7396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-1st 7678  df-2nd 7679
This theorem is referenced by:  ofpreima  30338  rrvadd  31609
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