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Theorem ofcfval2 30473
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 3102 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵𝑋)
3 eqid 2758 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 6179 . . . 4 (∀𝑥𝐴 𝐵𝑋 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 17 . . 3 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 ofcfval2.4 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 6140 . . 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 6453 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
128, 9, 10, 11ofcfval 30467 1 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1630  wcel 2137  wral 3048  cmpt 4879   Fn wfn 6042  (class class class)co 6811  𝑓/𝑐cofc 30464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-ofc 30465
This theorem is referenced by:  coinflippv  30852  ofcs1  30928
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