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Theorem ofcfval2 30506
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/𝑐𝑅𝐶) = (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 3115 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵𝑋)
3 eqid 2771 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 6159 . . . 4 (∀𝑥𝐴 𝐵𝑋 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 17 . . 3 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 ofcfval2.4 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 6120 . . 3 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 247 . 2 (𝜑𝐹 Fn 𝐴)
9 ofcfval2.1 . 2 (𝜑𝐴𝑉)
10 ofcfval2.2 . 2 (𝜑𝐶𝑊)
116, 1fvmpt2d 6437 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
128, 9, 10, 11ofcfval 30500 1 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  cmpt 4864   Fn wfn 6025  (class class class)co 6796  𝑓/𝑐cofc 30497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-ofc 30498
This theorem is referenced by:  coinflippv  30885  ofcs1  30961
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