Theorem pw2f1o2 43020

Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 9025, 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 43019	. 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 1540   ∈ wcel 2109  ∅c0 4292  ifcif 4484  𝒫 cpw 4559  {csn 4585   ↦ cmpt 5183  ^◡ccnv 5630   “ cima 5634  –1-1-onto→wf1o 6498  (class class class)co 7369  1_oc1o 8404  2_oc2o 8405   ↑_m cmap 8776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1o 8411  df-2o 8412  df-map 8778

This theorem is referenced by:  wepwsolem  43024  pwfi2f1o  43078