Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofoprabco Structured version   Visualization version   GIF version

Theorem ofoprabco 32495
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 7089 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐹𝑎) ∈ 𝐵)
4 ofoprabco.3 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
54ffvelcdmda 7089 . . . . . . 7 ((𝜑𝑎𝐴) → (𝐺𝑎) ∈ 𝐶)
6 opelxpi 5709 . . . . . . 7 (((𝐹𝑎) ∈ 𝐵 ∧ (𝐺𝑎) ∈ 𝐶) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (𝐵 × 𝐶))
73, 5, 6syl2anc 582 . . . . . 6 ((𝜑𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ (𝐵 × 𝐶))
81, 7fvmpt2d 7013 . . . . 5 ((𝜑𝑎𝐴) → (𝑀𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
98fveq2d 6896 . . . 4 ((𝜑𝑎𝐴) → (𝑁‘(𝑀𝑎)) = (𝑁‘⟨(𝐹𝑎), (𝐺𝑎)⟩))
10 df-ov 7419 . . . . 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 479 . . . . 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 7434 . . . . 5 (((𝜑𝑎𝐴) ∧ (𝑥 = (𝐹𝑎) ∧ 𝑦 = (𝐺𝑎))) → (𝑥𝑅𝑦) = ((𝐹𝑎)𝑅(𝐺𝑎)))
17 ovexd 7451 . . . . 5 ((𝜑𝑎𝐴) → ((𝐹𝑎)𝑅(𝐺𝑎)) ∈ V)
1813, 16, 3, 5, 17ovmpod 7570 . . . 4 ((𝜑𝑎𝐴) → ((𝐹𝑎)𝑁(𝐺𝑎)) = ((𝐹𝑎)𝑅(𝐺𝑎)))
199, 11, 183eqtr2d 2771 . . 3 ((𝜑𝑎𝐴) → (𝑁‘(𝑀𝑎)) = ((𝐹𝑎)𝑅(𝐺𝑎)))
2019mpteq2dva 5243 . 2 (𝜑 → (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎)𝑅(𝐺𝑎))))
21 ovex 7449 . . . . . 6 (𝑥𝑅𝑦) ∈ V
2221rgen2w 3056 . . . . 5 𝑥𝐵𝑦𝐶 (𝑥𝑅𝑦) ∈ V
23 eqid 2725 . . . . . 6 (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)) = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦))
2423fmpo 8070 . . . . 5 (∀𝑥𝐵𝑦𝐶 (𝑥𝑅𝑦) ∈ V ↔ (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V)
2522, 24mpbi 229 . . . 4 (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V
2612feq1d 6702 . . . 4 (𝜑 → (𝑁:(𝐵 × 𝐶)⟶V ↔ (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V))
2725, 26mpbiri 257 . . 3 (𝜑𝑁:(𝐵 × 𝐶)⟶V)
281, 7fmpt3d 7121 . . 3 (𝜑𝑀:𝐴⟶(𝐵 × 𝐶))
29 ofoprabco.1 . . . 4 𝑎𝑀
3029fcomptf 32489 . . 3 ((𝑁:(𝐵 × 𝐶)⟶V ∧ 𝑀:𝐴⟶(𝐵 × 𝐶)) → (𝑁𝑀) = (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))))
3127, 28, 30syl2anc 582 . 2 (𝜑 → (𝑁𝑀) = (𝑎𝐴 ↦ (𝑁‘(𝑀𝑎))))
32 ofoprabco.4 . . 3 (𝜑𝐴𝑉)
332feqmptd 6962 . . 3 (𝜑𝐹 = (𝑎𝐴 ↦ (𝐹𝑎)))
344feqmptd 6962 . . 3 (𝜑𝐺 = (𝑎𝐴 ↦ (𝐺𝑎)))
3532, 3, 5, 33, 34offval2 7702 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎)𝑅(𝐺𝑎))))
3620, 31, 353eqtr4rd 2776 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝑁𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wnfc 2875  wral 3051  Vcvv 3463  cop 4630  cmpt 5226   × cxp 5670  ccom 5676  wf 6539  cfv 6543  (class class class)co 7416  cmpo 7418  f cof 7680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-1st 7991  df-2nd 7992
This theorem is referenced by:  ofpreima  32496  rrvadd  34129
  Copyright terms: Public domain W3C validator