Theorem pw2f1o2 38125

Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8234, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypothesis

Ref	Expression
pw2f1o2.f	⊢ 𝐹 = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜}))

Assertion

Ref	Expression
pw2f1o2	⊢ (𝐴 ∈ 𝑉 → 𝐹:(2𝑜 ↑𝑚 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.

Step	Hyp	Ref	Expression
1		pw2f1o2.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜}))
2	1	pw2f1ocnv 38124	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹:(2𝑜 ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴 ∧ ◡𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)))))
3	2	simpld 477	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:(2𝑜 ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)

Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  ∅c0 4058  ifcif 4230  𝒫 cpw 4302  {csn 4321   ↦ cmpt 4881  ^◡ccnv 5265   “ cima 5269  –1-1-onto→wf1o 6048  (class class class)co 6814  1_𝑜c1o 7723  2_𝑜c2o 7724   ↑_𝑚 cmap 8025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1o 7730  df-2o 7731  df-map 8027

This theorem is referenced by:  wepwsolem  38132  pwfi2f1o  38186