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Theorem ofcfeqd2 31036
 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/𝑐𝑅𝐶) = (𝐹𝑓/𝑐𝑃𝐶))
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 6981 . . . . 5 (𝑦 = (𝐹𝑥) → (𝑦𝑅𝐶) = ((𝐹𝑥)𝑅𝐶))
2 oveq1 6981 . . . . 5 (𝑦 = (𝐹𝑥) → (𝑦𝑃𝐶) = ((𝐹𝑥)𝑃𝐶))
31, 2eqeq12d 2786 . . . 4 (𝑦 = (𝐹𝑥) → ((𝑦𝑅𝐶) = (𝑦𝑃𝐶) ↔ ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑥)𝑃𝐶)))
4 ofcfeqd2.2 . . . . . 6 ((𝜑𝑦𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
54ralrimiva 3125 . . . . 5 (𝜑 → ∀𝑦𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
65adantr 473 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
7 ofcfeqd2.1 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
83, 6, 7rspcdva 3534 . . 3 ((𝜑𝑥𝐴) → ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑥)𝑃𝐶))
98mpteq2dva 5018 . 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 2772 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
1410, 11, 12, 13ofcfval 31033 . 2 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
1510, 11, 12, 13ofcfval 31033 . 2 (𝜑 → (𝐹𝑓/𝑐𝑃𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑃𝐶)))
169, 14, 153eqtr4d 2817 1 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝐹𝑓/𝑐𝑃𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051  ∀wral 3081   ↦ cmpt 5004   Fn wfn 6180  ‘cfv 6185  (class class class)co 6974  ∘𝑓/𝑐cofc 31030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pr 5182 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-ofc 31031 This theorem is referenced by:  coinfliplem  31414
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