Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofcfeqd2 Structured version   Visualization version   GIF version

Theorem ofcfeqd2 33628
Description: Equality theorem for function/constant operation value. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcfeqd2.1 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
ofcfeqd2.2 ((𝜑𝑦𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
ofcfeqd2.3 (𝜑𝐹 Fn 𝐴)
ofcfeqd2.4 (𝜑𝐴𝑉)
ofcfeqd2.5 (𝜑𝐶𝑊)
Assertion
Ref Expression
ofcfeqd2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝐹f/c 𝑃𝐶))
Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐹,𝑦   𝑥,𝑃,𝑦   𝑥,𝑅,𝑦   𝜑,𝑥,𝑦   𝑦,𝐵
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem ofcfeqd2
StepHypRef Expression
1 oveq1 7411 . . . . 5 (𝑦 = (𝐹𝑥) → (𝑦𝑅𝐶) = ((𝐹𝑥)𝑅𝐶))
2 oveq1 7411 . . . . 5 (𝑦 = (𝐹𝑥) → (𝑦𝑃𝐶) = ((𝐹𝑥)𝑃𝐶))
31, 2eqeq12d 2742 . . . 4 (𝑦 = (𝐹𝑥) → ((𝑦𝑅𝐶) = (𝑦𝑃𝐶) ↔ ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑥)𝑃𝐶)))
4 ofcfeqd2.2 . . . . . 6 ((𝜑𝑦𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
54ralrimiva 3140 . . . . 5 (𝜑 → ∀𝑦𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
65adantr 480 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
7 ofcfeqd2.1 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
83, 6, 7rspcdva 3607 . . 3 ((𝜑𝑥𝐴) → ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑥)𝑃𝐶))
98mpteq2dva 5241 . 2 (𝜑 → (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑃𝐶)))
10 ofcfeqd2.3 . . 3 (𝜑𝐹 Fn 𝐴)
11 ofcfeqd2.4 . . 3 (𝜑𝐴𝑉)
12 ofcfeqd2.5 . . 3 (𝜑𝐶𝑊)
13 eqidd 2727 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
1410, 11, 12, 13ofcfval 33625 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
1510, 11, 12, 13ofcfval 33625 . 2 (𝜑 → (𝐹f/c 𝑃𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑃𝐶)))
169, 14, 153eqtr4d 2776 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝐹f/c 𝑃𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  cmpt 5224   Fn wfn 6531  cfv 6536  (class class class)co 7404  f/c cofc 33622
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-ofc 33623
This theorem is referenced by:  coinfliplem  34006
  Copyright terms: Public domain W3C validator