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Theorem ofcfeqd2 33947
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 7423 . . . . 5 (𝑦 = (𝐹𝑥) → (𝑦𝑅𝐶) = ((𝐹𝑥)𝑅𝐶))
2 oveq1 7423 . . . . 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 3136 . . . . 5 (𝜑 → ∀𝑦𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
65adantr 479 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
7 ofcfeqd2.1 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
83, 6, 7rspcdva 3608 . . 3 ((𝜑𝑥𝐴) → ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑥)𝑃𝐶))
98mpteq2dva 5245 . 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 33944 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
1510, 11, 12, 13ofcfval 33944 . 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 394   = wceq 1534  wcel 2099  wral 3051  cmpt 5228   Fn wfn 6541  cfv 6546  (class class class)co 7416  f/c cofc 33941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-ofc 33942
This theorem is referenced by:  coinfliplem  34325
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