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Theorem pw2f1ocnv 36622
 Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7952, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 9-Jul-2015.)
Hypothesis
Ref Expression
pw2f1o2.f 𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))
Assertion
Ref Expression
pw2f1ocnv (𝐴𝑉 → (𝐹:(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝑉,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧)   𝑉(𝑧)

Proof of Theorem pw2f1ocnv
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 pw2f1o2.f . 2 𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))
2 vex 3176 . . . 4 𝑥 ∈ V
32cnvex 7006 . . 3 𝑥 ∈ V
4 imaexg 6995 . . 3 (𝑥 ∈ V → (𝑥 “ {1𝑜}) ∈ V)
53, 4mp1i 13 . 2 ((𝐴𝑉𝑥 ∈ (2𝑜𝑚 𝐴)) → (𝑥 “ {1𝑜}) ∈ V)
6 mptexg 6389 . . 3 (𝐴𝑉 → (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ∈ V)
76adantr 480 . 2 ((𝐴𝑉𝑦 ∈ 𝒫 𝐴) → (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ∈ V)
8 2on 7455 . . . . . 6 2𝑜 ∈ On
9 elmapg 7757 . . . . . 6 ((2𝑜 ∈ On ∧ 𝐴𝑉) → (𝑥 ∈ (2𝑜𝑚 𝐴) ↔ 𝑥:𝐴⟶2𝑜))
108, 9mpan 702 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (2𝑜𝑚 𝐴) ↔ 𝑥:𝐴⟶2𝑜))
1110anbi1d 737 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1𝑜})) ↔ (𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜}))))
12 1on 7454 . . . . . . . . . . . . 13 1𝑜 ∈ On
1312elexi 3186 . . . . . . . . . . . 12 1𝑜 ∈ V
1413sucid 5721 . . . . . . . . . . 11 1𝑜 ∈ suc 1𝑜
15 df-2o 7448 . . . . . . . . . . 11 2𝑜 = suc 1𝑜
1614, 15eleqtrri 2687 . . . . . . . . . 10 1𝑜 ∈ 2𝑜
17 0ex 4718 . . . . . . . . . . . 12 ∅ ∈ V
1817prid1 4241 . . . . . . . . . . 11 ∅ ∈ {∅, {∅}}
19 df2o2 7461 . . . . . . . . . . 11 2𝑜 = {∅, {∅}}
2018, 19eleqtrri 2687 . . . . . . . . . 10 ∅ ∈ 2𝑜
2116, 20keepel 4105 . . . . . . . . 9 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜
2221rgenw 2908 . . . . . . . 8 𝑧𝐴 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜
23 eqid 2610 . . . . . . . . 9 (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))
2423fmpt 6289 . . . . . . . 8 (∀𝑧𝐴 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜 ↔ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜)
2522, 24mpbi 219 . . . . . . 7 (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜
26 simpr 476 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))
2726feq1d 5943 . . . . . . 7 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑥:𝐴⟶2𝑜 ↔ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜))
2825, 27mpbiri 247 . . . . . 6 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → 𝑥:𝐴⟶2𝑜)
2926fveq1d 6105 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
30 elequ1 1984 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧𝑦𝑤𝑦))
3130ifbid 4058 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → if(𝑧𝑦, 1𝑜, ∅) = if(𝑤𝑦, 1𝑜, ∅))
3213, 17keepel 4105 . . . . . . . . . . . . . 14 if(𝑤𝑦, 1𝑜, ∅) ∈ V
3331, 23, 32fvmpt 6191 . . . . . . . . . . . . 13 (𝑤𝐴 → ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤) = if(𝑤𝑦, 1𝑜, ∅))
3429, 33sylan9eq 2664 . . . . . . . . . . . 12 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
3534eqeq1d 2612 . . . . . . . . . . 11 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1𝑜 ↔ if(𝑤𝑦, 1𝑜, ∅) = 1𝑜))
36 iftrue 4042 . . . . . . . . . . . 12 (𝑤𝑦 → if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
37 noel 3878 . . . . . . . . . . . . . 14 ¬ ∅ ∈ ∅
38 iffalse 4045 . . . . . . . . . . . . . . . 16 𝑤𝑦 → if(𝑤𝑦, 1𝑜, ∅) = ∅)
3938eqeq1d 2612 . . . . . . . . . . . . . . 15 𝑤𝑦 → (if(𝑤𝑦, 1𝑜, ∅) = 1𝑜 ↔ ∅ = 1𝑜))
40 0lt1o 7471 . . . . . . . . . . . . . . . 16 ∅ ∈ 1𝑜
41 eleq2 2677 . . . . . . . . . . . . . . . 16 (∅ = 1𝑜 → (∅ ∈ ∅ ↔ ∅ ∈ 1𝑜))
4240, 41mpbiri 247 . . . . . . . . . . . . . . 15 (∅ = 1𝑜 → ∅ ∈ ∅)
4339, 42syl6bi 242 . . . . . . . . . . . . . 14 𝑤𝑦 → (if(𝑤𝑦, 1𝑜, ∅) = 1𝑜 → ∅ ∈ ∅))
4437, 43mtoi 189 . . . . . . . . . . . . 13 𝑤𝑦 → ¬ if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
4544con4i 112 . . . . . . . . . . . 12 (if(𝑤𝑦, 1𝑜, ∅) = 1𝑜𝑤𝑦)
4636, 45impbii 198 . . . . . . . . . . 11 (𝑤𝑦 ↔ if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
4735, 46syl6rbbr 278 . . . . . . . . . 10 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) = 1𝑜))
48 fvex 6113 . . . . . . . . . . 11 (𝑥𝑤) ∈ V
4948elsn 4140 . . . . . . . . . 10 ((𝑥𝑤) ∈ {1𝑜} ↔ (𝑥𝑤) = 1𝑜)
5047, 49syl6bbr 277 . . . . . . . . 9 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) ∈ {1𝑜}))
5150pm5.32da 671 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → ((𝑤𝐴𝑤𝑦) ↔ (𝑤𝐴 ∧ (𝑥𝑤) ∈ {1𝑜})))
52 ssel 3562 . . . . . . . . . 10 (𝑦𝐴 → (𝑤𝑦𝑤𝐴))
5352adantr 480 . . . . . . . . 9 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤𝑦𝑤𝐴))
5453pm4.71rd 665 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤𝑦 ↔ (𝑤𝐴𝑤𝑦)))
55 ffn 5958 . . . . . . . . 9 (𝑥:𝐴⟶2𝑜𝑥 Fn 𝐴)
56 elpreima 6245 . . . . . . . . 9 (𝑥 Fn 𝐴 → (𝑤 ∈ (𝑥 “ {1𝑜}) ↔ (𝑤𝐴 ∧ (𝑥𝑤) ∈ {1𝑜})))
5728, 55, 563syl 18 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤 ∈ (𝑥 “ {1𝑜}) ↔ (𝑤𝐴 ∧ (𝑥𝑤) ∈ {1𝑜})))
5851, 54, 573bitr4d 299 . . . . . . 7 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤𝑦𝑤 ∈ (𝑥 “ {1𝑜})))
5958eqrdv 2608 . . . . . 6 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → 𝑦 = (𝑥 “ {1𝑜}))
6028, 59jca 553 . . . . 5 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})))
61 simpr 476 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑦 = (𝑥 “ {1𝑜}))
62 cnvimass 5404 . . . . . . . 8 (𝑥 “ {1𝑜}) ⊆ dom 𝑥
63 fdm 5964 . . . . . . . . 9 (𝑥:𝐴⟶2𝑜 → dom 𝑥 = 𝐴)
6463adantr 480 . . . . . . . 8 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → dom 𝑥 = 𝐴)
6562, 64syl5sseq 3616 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → (𝑥 “ {1𝑜}) ⊆ 𝐴)
6661, 65eqsstrd 3602 . . . . . 6 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑦𝐴)
67 simplr 788 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → 𝑦 = (𝑥 “ {1𝑜}))
6867eleq2d 2673 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑤𝑦𝑤 ∈ (𝑥 “ {1𝑜})))
6955adantr 480 . . . . . . . . . . . . . . 15 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑥 Fn 𝐴)
70 fnbrfvb 6146 . . . . . . . . . . . . . . 15 ((𝑥 Fn 𝐴𝑤𝐴) → ((𝑥𝑤) = 1𝑜𝑤𝑥1𝑜))
7169, 70sylan 487 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1𝑜𝑤𝑥1𝑜))
72 vex 3176 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
7372eliniseg 5413 . . . . . . . . . . . . . . 15 (1𝑜 ∈ On → (𝑤 ∈ (𝑥 “ {1𝑜}) ↔ 𝑤𝑥1𝑜))
7412, 73ax-mp 5 . . . . . . . . . . . . . 14 (𝑤 ∈ (𝑥 “ {1𝑜}) ↔ 𝑤𝑥1𝑜)
7571, 74syl6bbr 277 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1𝑜𝑤 ∈ (𝑥 “ {1𝑜})))
7668, 75bitr4d 270 . . . . . . . . . . . 12 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) = 1𝑜))
7776biimpa 500 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = 1𝑜)
7836adantl 481 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
7977, 78eqtr4d 2647 . . . . . . . . . 10 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
80 ffvelrn 6265 . . . . . . . . . . . . . . . . . 18 ((𝑥:𝐴⟶2𝑜𝑤𝐴) → (𝑥𝑤) ∈ 2𝑜)
8180adantlr 747 . . . . . . . . . . . . . . . . 17 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ 2𝑜)
82 df2o3 7460 . . . . . . . . . . . . . . . . 17 2𝑜 = {∅, 1𝑜}
8381, 82syl6eleq 2698 . . . . . . . . . . . . . . . 16 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ {∅, 1𝑜})
8448elpr 4146 . . . . . . . . . . . . . . . 16 ((𝑥𝑤) ∈ {∅, 1𝑜} ↔ ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1𝑜))
8583, 84sylib 207 . . . . . . . . . . . . . . 15 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1𝑜))
8685ord 391 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (¬ (𝑥𝑤) = ∅ → (𝑥𝑤) = 1𝑜))
8786, 76sylibrd 248 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (¬ (𝑥𝑤) = ∅ → 𝑤𝑦))
8887con1d 138 . . . . . . . . . . . 12 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (¬ 𝑤𝑦 → (𝑥𝑤) = ∅))
8988imp 444 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = ∅)
9038adantl 481 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → if(𝑤𝑦, 1𝑜, ∅) = ∅)
9189, 90eqtr4d 2647 . . . . . . . . . 10 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9279, 91pm2.61dan 828 . . . . . . . . 9 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9333adantl 481 . . . . . . . . 9 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9492, 93eqtr4d 2647 . . . . . . . 8 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
9594ralrimiva 2949 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
96 ffn 5958 . . . . . . . . 9 ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜 → (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) Fn 𝐴)
9725, 96ax-mp 5 . . . . . . . 8 (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) Fn 𝐴
98 eqfnfv 6219 . . . . . . . 8 ((𝑥 Fn 𝐴 ∧ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) Fn 𝐴) → (𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ↔ ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤)))
9969, 97, 98sylancl 693 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → (𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ↔ ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤)))
10095, 99mpbird 246 . . . . . 6 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))
10166, 100jca 553 . . . . 5 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))))
10260, 101impbii 198 . . . 4 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ↔ (𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})))
10311, 102syl6bbr 277 . . 3 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1𝑜})) ↔ (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))))
104 selpw 4115 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
105104anbi1i 727 . . 3 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ↔ (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))))
106103, 105syl6bbr 277 . 2 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1𝑜})) ↔ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))))
1071, 5, 7, 106f1ocnvd 6782 1 (𝐴𝑉 → (𝐹:(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038   “ cima 5041  Oncon0 5640  suc csuc 5642   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1𝑜c1o 7440  2𝑜c2o 7441   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-2o 7448  df-map 7746 This theorem is referenced by:  pw2f1o2  36623
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