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Theorem ofcfval2 33632
Description: The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcfval2.1 (𝜑𝐴𝑉)
ofcfval2.2 (𝜑𝐶𝑊)
ofcfval2.3 ((𝜑𝑥𝐴) → 𝐵𝑋)
ofcfval2.4 (𝜑𝐹 = (𝑥𝐴𝐵))
Assertion
Ref Expression
ofcfval2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹   𝑥,𝑅   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem ofcfval2
StepHypRef Expression
1 ofcfval2.3 . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑋)
21ralrimiva 3140 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵𝑋)
3 eqid 2726 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 6684 . . . 4 (∀𝑥𝐴 𝐵𝑋 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 17 . . 3 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 ofcfval2.4 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 6636 . . 3 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 257 . 2 (𝜑𝐹 Fn 𝐴)
9 ofcfval2.1 . 2 (𝜑𝐴𝑉)
10 ofcfval2.2 . 2 (𝜑𝐶𝑊)
116, 1fvmpt2d 7005 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
128, 9, 10, 11ofcfval 33626 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  cmpt 5224   Fn wfn 6532  (class class class)co 7405  f/c cofc 33623
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 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-ofc 33624
This theorem is referenced by:  coinflippv  34012  ofcs1  34085
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