Theorem pw2f1o2 43481

Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 9013, 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 43480	. 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 494	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:(2o ↑m 𝐴)–1-1-onto→𝒫 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∅c0 4274  ifcif 4467  𝒫 cpw 4542  {csn 4568   ↦ cmpt 5167  ^◡ccnv 5621   “ cima 5625  –1-1-onto→wf1o 6489  (class class class)co 7358  1_oc1o 8389  2_oc2o 8390   ↑_m cmap 8764

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1o 8396  df-2o 8397  df-map 8766

This theorem is referenced by:  wepwsolem  43485  pwfi2f1o  43539