pw2f1o2 - Mathbox for Stefan O'Rear

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Theorem pw2f1o2 43983

Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 9081, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)

**Hypothesis**
Ref	Expression
pw2f1o2.f	⊢ 𝐹 = (𝑥 ∈ (2_o ↑_m 𝐴) ↦ (^◡𝑥 “ {1_o}))

**Assertion**
Ref	Expression
pw2f1o2	⊢ (𝐴 ∈ 𝑉 → 𝐹:(2_o ↑_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 ∈ (2_o ↑_m 𝐴) ↦ (^◡𝑥 “ {1_o}))
2	1	pw2f1ocnv 43982	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹:(2_o ↑_m 𝐴)–1-1-onto→𝒫 𝐴 ∧ ^◡𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1_o, ∅)))))
3	2	simpld 500	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:(2_o ↑_m 𝐴)–1-1-onto→𝒫 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ∅c0 4278 ifcif 4481 𝒫 cpw 4556 {csn 4583 ↦ cmpt 5185 ^◡ccnv 5646 “ cima 5650 –1-1-onto→wf1o 6526 (class class class)co 7408 1_oc1o 8447 2_oc2o 8448 ↑_m cmap 8825

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-ord 6354 df-on 6355 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1o 8454 df-2o 8455 df-map 8827

This theorem is used by: wepwsolem 43987 pwfi2f1o 44041

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