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Theorem ofcfval2 33090
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 3146 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵𝑋)
3 eqid 2732 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 6687 . . . 4 (∀𝑥𝐴 𝐵𝑋 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 17 . . 3 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 ofcfval2.4 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 6639 . . 3 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 256 . 2 (𝜑𝐹 Fn 𝐴)
9 ofcfval2.1 . 2 (𝜑𝐴𝑉)
10 ofcfval2.2 . 2 (𝜑𝐶𝑊)
116, 1fvmpt2d 7008 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
128, 9, 10, 11ofcfval 33084 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  cmpt 5230   Fn wfn 6535  (class class class)co 7405  f/c cofc 33081
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-ofc 33082
This theorem is referenced by:  coinflippv  33470  ofcs1  33543
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