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Theorem pw2f1ocnv 38106
Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8232, 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 3343 . . . 4 𝑥 ∈ V
32cnvex 7278 . . 3 𝑥 ∈ V
4 imaexg 7268 . . 3 (𝑥 ∈ V → (𝑥 “ {1𝑜}) ∈ V)
53, 4mp1i 13 . 2 ((𝐴𝑉𝑥 ∈ (2𝑜𝑚 𝐴)) → (𝑥 “ {1𝑜}) ∈ V)
6 mptexg 6648 . . 3 (𝐴𝑉 → (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ∈ V)
76adantr 472 . 2 ((𝐴𝑉𝑦 ∈ 𝒫 𝐴) → (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ∈ V)
8 2on 7737 . . . . . 6 2𝑜 ∈ On
9 elmapg 8036 . . . . . 6 ((2𝑜 ∈ On ∧ 𝐴𝑉) → (𝑥 ∈ (2𝑜𝑚 𝐴) ↔ 𝑥:𝐴⟶2𝑜))
108, 9mpan 708 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (2𝑜𝑚 𝐴) ↔ 𝑥:𝐴⟶2𝑜))
1110anbi1d 743 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1𝑜})) ↔ (𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜}))))
12 1on 7736 . . . . . . . . . . . . 13 1𝑜 ∈ On
1312elexi 3353 . . . . . . . . . . . 12 1𝑜 ∈ V
1413sucid 5965 . . . . . . . . . . 11 1𝑜 ∈ suc 1𝑜
15 df-2o 7730 . . . . . . . . . . 11 2𝑜 = suc 1𝑜
1614, 15eleqtrri 2838 . . . . . . . . . 10 1𝑜 ∈ 2𝑜
17 0ex 4942 . . . . . . . . . . . 12 ∅ ∈ V
1817prid1 4441 . . . . . . . . . . 11 ∅ ∈ {∅, {∅}}
19 df2o2 7743 . . . . . . . . . . 11 2𝑜 = {∅, {∅}}
2018, 19eleqtrri 2838 . . . . . . . . . 10 ∅ ∈ 2𝑜
2116, 20keepel 4299 . . . . . . . . 9 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜
2221rgenw 3062 . . . . . . . 8 𝑧𝐴 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜
23 eqid 2760 . . . . . . . . 9 (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))
2423fmpt 6544 . . . . . . . 8 (∀𝑧𝐴 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜 ↔ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜)
2522, 24mpbi 220 . . . . . . 7 (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜
26 simpr 479 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))
2726feq1d 6191 . . . . . . 7 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑥:𝐴⟶2𝑜 ↔ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜))
2825, 27mpbiri 248 . . . . . 6 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → 𝑥:𝐴⟶2𝑜)
2926fveq1d 6354 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
30 elequ1 2146 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧𝑦𝑤𝑦))
3130ifbid 4252 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → if(𝑧𝑦, 1𝑜, ∅) = if(𝑤𝑦, 1𝑜, ∅))
3213, 17keepel 4299 . . . . . . . . . . . . . 14 if(𝑤𝑦, 1𝑜, ∅) ∈ V
3331, 23, 32fvmpt 6444 . . . . . . . . . . . . 13 (𝑤𝐴 → ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤) = if(𝑤𝑦, 1𝑜, ∅))
3429, 33sylan9eq 2814 . . . . . . . . . . . 12 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
3534eqeq1d 2762 . . . . . . . . . . 11 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1𝑜 ↔ if(𝑤𝑦, 1𝑜, ∅) = 1𝑜))
36 iftrue 4236 . . . . . . . . . . . 12 (𝑤𝑦 → if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
37 noel 4062 . . . . . . . . . . . . . 14 ¬ ∅ ∈ ∅
38 iffalse 4239 . . . . . . . . . . . . . . . 16 𝑤𝑦 → if(𝑤𝑦, 1𝑜, ∅) = ∅)
3938eqeq1d 2762 . . . . . . . . . . . . . . 15 𝑤𝑦 → (if(𝑤𝑦, 1𝑜, ∅) = 1𝑜 ↔ ∅ = 1𝑜))
40 0lt1o 7753 . . . . . . . . . . . . . . . 16 ∅ ∈ 1𝑜
41 eleq2 2828 . . . . . . . . . . . . . . . 16 (∅ = 1𝑜 → (∅ ∈ ∅ ↔ ∅ ∈ 1𝑜))
4240, 41mpbiri 248 . . . . . . . . . . . . . . 15 (∅ = 1𝑜 → ∅ ∈ ∅)
4339, 42syl6bi 243 . . . . . . . . . . . . . 14 𝑤𝑦 → (if(𝑤𝑦, 1𝑜, ∅) = 1𝑜 → ∅ ∈ ∅))
4437, 43mtoi 190 . . . . . . . . . . . . 13 𝑤𝑦 → ¬ if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
4544con4i 113 . . . . . . . . . . . 12 (if(𝑤𝑦, 1𝑜, ∅) = 1𝑜𝑤𝑦)
4636, 45impbii 199 . . . . . . . . . . 11 (𝑤𝑦 ↔ if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
4735, 46syl6rbbr 279 . . . . . . . . . 10 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) = 1𝑜))
48 fvex 6362 . . . . . . . . . . 11 (𝑥𝑤) ∈ V
4948elsn 4336 . . . . . . . . . 10 ((𝑥𝑤) ∈ {1𝑜} ↔ (𝑥𝑤) = 1𝑜)
5047, 49syl6bbr 278 . . . . . . . . 9 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) ∈ {1𝑜}))
5150pm5.32da 676 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → ((𝑤𝐴𝑤𝑦) ↔ (𝑤𝐴 ∧ (𝑥𝑤) ∈ {1𝑜})))
52 ssel 3738 . . . . . . . . . 10 (𝑦𝐴 → (𝑤𝑦𝑤𝐴))
5352adantr 472 . . . . . . . . 9 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤𝑦𝑤𝐴))
5453pm4.71rd 670 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤𝑦 ↔ (𝑤𝐴𝑤𝑦)))
55 ffn 6206 . . . . . . . . 9 (𝑥:𝐴⟶2𝑜𝑥 Fn 𝐴)
56 elpreima 6500 . . . . . . . . 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 300 . . . . . . 7 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤𝑦𝑤 ∈ (𝑥 “ {1𝑜})))
5958eqrdv 2758 . . . . . 6 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → 𝑦 = (𝑥 “ {1𝑜}))
6028, 59jca 555 . . . . 5 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})))
61 simpr 479 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑦 = (𝑥 “ {1𝑜}))
62 cnvimass 5643 . . . . . . . 8 (𝑥 “ {1𝑜}) ⊆ dom 𝑥
63 fdm 6212 . . . . . . . . 9 (𝑥:𝐴⟶2𝑜 → dom 𝑥 = 𝐴)
6463adantr 472 . . . . . . . 8 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → dom 𝑥 = 𝐴)
6562, 64syl5sseq 3794 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → (𝑥 “ {1𝑜}) ⊆ 𝐴)
6661, 65eqsstrd 3780 . . . . . 6 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑦𝐴)
67 simplr 809 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → 𝑦 = (𝑥 “ {1𝑜}))
6867eleq2d 2825 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑤𝑦𝑤 ∈ (𝑥 “ {1𝑜})))
6955adantr 472 . . . . . . . . . . . . . . 15 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑥 Fn 𝐴)
70 fnbrfvb 6397 . . . . . . . . . . . . . . 15 ((𝑥 Fn 𝐴𝑤𝐴) → ((𝑥𝑤) = 1𝑜𝑤𝑥1𝑜))
7169, 70sylan 489 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1𝑜𝑤𝑥1𝑜))
72 vex 3343 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
7372eliniseg 5652 . . . . . . . . . . . . . . 15 (1𝑜 ∈ On → (𝑤 ∈ (𝑥 “ {1𝑜}) ↔ 𝑤𝑥1𝑜))
7412, 73ax-mp 5 . . . . . . . . . . . . . 14 (𝑤 ∈ (𝑥 “ {1𝑜}) ↔ 𝑤𝑥1𝑜)
7571, 74syl6bbr 278 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1𝑜𝑤 ∈ (𝑥 “ {1𝑜})))
7668, 75bitr4d 271 . . . . . . . . . . . 12 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) = 1𝑜))
7776biimpa 502 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = 1𝑜)
7836adantl 473 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
7977, 78eqtr4d 2797 . . . . . . . . . 10 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
80 ffvelrn 6520 . . . . . . . . . . . . . . . . . 18 ((𝑥:𝐴⟶2𝑜𝑤𝐴) → (𝑥𝑤) ∈ 2𝑜)
8180adantlr 753 . . . . . . . . . . . . . . . . 17 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ 2𝑜)
82 df2o3 7742 . . . . . . . . . . . . . . . . 17 2𝑜 = {∅, 1𝑜}
8381, 82syl6eleq 2849 . . . . . . . . . . . . . . . 16 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ {∅, 1𝑜})
8448elpr 4343 . . . . . . . . . . . . . . . 16 ((𝑥𝑤) ∈ {∅, 1𝑜} ↔ ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1𝑜))
8583, 84sylib 208 . . . . . . . . . . . . . . 15 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1𝑜))
8685ord 391 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (¬ (𝑥𝑤) = ∅ → (𝑥𝑤) = 1𝑜))
8786, 76sylibrd 249 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (¬ (𝑥𝑤) = ∅ → 𝑤𝑦))
8887con1d 139 . . . . . . . . . . . 12 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (¬ 𝑤𝑦 → (𝑥𝑤) = ∅))
8988imp 444 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = ∅)
9038adantl 473 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → if(𝑤𝑦, 1𝑜, ∅) = ∅)
9189, 90eqtr4d 2797 . . . . . . . . . 10 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9279, 91pm2.61dan 867 . . . . . . . . 9 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9333adantl 473 . . . . . . . . 9 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9492, 93eqtr4d 2797 . . . . . . . 8 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
9594ralrimiva 3104 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
96 ffn 6206 . . . . . . . . 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 6474 . . . . . . . 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 697 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → (𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ↔ ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤)))
10095, 99mpbird 247 . . . . . 6 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))
10166, 100jca 555 . . . . 5 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))))
10260, 101impbii 199 . . . 4 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ↔ (𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})))
10311, 102syl6bbr 278 . . 3 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1𝑜})) ↔ (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))))
104 selpw 4309 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
105104anbi1i 733 . . 3 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ↔ (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))))
106103, 105syl6bbr 278 . 2 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1𝑜})) ↔ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))))
1071, 5, 7, 106f1ocnvd 7049 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 196  wo 382  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715  c0 4058  ifcif 4230  𝒫 cpw 4302  {csn 4321  {cpr 4323   class class class wbr 4804  cmpt 4881  ccnv 5265  dom cdm 5266  cima 5269  Oncon0 5884  suc csuc 5886   Fn wfn 6044  wf 6045  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  1𝑜c1o 7722  2𝑜c2o 7723  𝑚 cmap 8023
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 7114
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 6816  df-oprab 6817  df-mpt2 6818  df-1o 7729  df-2o 7730  df-map 8025
This theorem is referenced by:  pw2f1o2  38107
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