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Theorem ofcfeqd2 31434
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 7147 . . . . 5 (𝑦 = (𝐹𝑥) → (𝑦𝑅𝐶) = ((𝐹𝑥)𝑅𝐶))
2 oveq1 7147 . . . . 5 (𝑦 = (𝐹𝑥) → (𝑦𝑃𝐶) = ((𝐹𝑥)𝑃𝐶))
31, 2eqeq12d 2838 . . . 4 (𝑦 = (𝐹𝑥) → ((𝑦𝑅𝐶) = (𝑦𝑃𝐶) ↔ ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑥)𝑃𝐶)))
4 ofcfeqd2.2 . . . . . 6 ((𝜑𝑦𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
54ralrimiva 3174 . . . . 5 (𝜑 → ∀𝑦𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
65adantr 484 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
7 ofcfeqd2.1 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
83, 6, 7rspcdva 3600 . . 3 ((𝜑𝑥𝐴) → ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑥)𝑃𝐶))
98mpteq2dva 5137 . 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 2823 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
1410, 11, 12, 13ofcfval 31431 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
1510, 11, 12, 13ofcfval 31431 . 2 (𝜑 → (𝐹f/c 𝑃𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑃𝐶)))
169, 14, 153eqtr4d 2867 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝐹f/c 𝑃𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  cmpt 5122   Fn wfn 6329  cfv 6334  (class class class)co 7140  f/c cofc 31428
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-ofc 31429
This theorem is referenced by:  coinfliplem  31810
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