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Theorem ofcfeqd2 31259
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 7152 . . . . 5 (𝑦 = (𝐹𝑥) → (𝑦𝑅𝐶) = ((𝐹𝑥)𝑅𝐶))
2 oveq1 7152 . . . . 5 (𝑦 = (𝐹𝑥) → (𝑦𝑃𝐶) = ((𝐹𝑥)𝑃𝐶))
31, 2eqeq12d 2834 . . . 4 (𝑦 = (𝐹𝑥) → ((𝑦𝑅𝐶) = (𝑦𝑃𝐶) ↔ ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑥)𝑃𝐶)))
4 ofcfeqd2.2 . . . . . 6 ((𝜑𝑦𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
54ralrimiva 3179 . . . . 5 (𝜑 → ∀𝑦𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
65adantr 481 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
7 ofcfeqd2.1 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
83, 6, 7rspcdva 3622 . . 3 ((𝜑𝑥𝐴) → ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑥)𝑃𝐶))
98mpteq2dva 5152 . 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 2819 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
1410, 11, 12, 13ofcfval 31256 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
1510, 11, 12, 13ofcfval 31256 . 2 (𝜑 → (𝐹f/c 𝑃𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑃𝐶)))
169, 14, 153eqtr4d 2863 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝐹f/c 𝑃𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  cmpt 5137   Fn wfn 6343  cfv 6348  (class class class)co 7145  f/c cofc 31253
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-pr 5320
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-ofc 31254
This theorem is referenced by:  coinfliplem  31635
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