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Theorem pw2f1o2 38564
 Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8355, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
pw2f1o2.f 𝐹 = (𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o}))
Assertion
Ref Expression
pw2f1o2 (𝐴𝑉𝐹:(2o𝑚 𝐴)–1-1-onto→𝒫 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem pw2f1o2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pw2f1o2.f . . 3 𝐹 = (𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o}))
21pw2f1ocnv 38563 . 2 (𝐴𝑉 → (𝐹:(2o𝑚 𝐴)–1-1-onto→𝒫 𝐴𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)))))
32simpld 490 1 (𝐴𝑉𝐹:(2o𝑚 𝐴)–1-1-onto→𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  ∅c0 4141  ifcif 4307  𝒫 cpw 4379  {csn 4398   ↦ cmpt 4965  ◡ccnv 5354   “ cima 5358  –1-1-onto→wf1o 6134  (class class class)co 6922  1oc1o 7836  2oc2o 7837   ↑𝑚 cmap 8140 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1o 7843  df-2o 7844  df-map 8142 This theorem is referenced by:  wepwsolem  38571  pwfi2f1o  38625
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