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Theorem pw2f1ocnv 37084
Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8011, 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 3189 . . . 4 𝑥 ∈ V
32cnvex 7060 . . 3 𝑥 ∈ V
4 imaexg 7050 . . 3 (𝑥 ∈ V → (𝑥 “ {1𝑜}) ∈ V)
53, 4mp1i 13 . 2 ((𝐴𝑉𝑥 ∈ (2𝑜𝑚 𝐴)) → (𝑥 “ {1𝑜}) ∈ V)
6 mptexg 6438 . . 3 (𝐴𝑉 → (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ∈ V)
76adantr 481 . 2 ((𝐴𝑉𝑦 ∈ 𝒫 𝐴) → (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ∈ V)
8 2on 7513 . . . . . 6 2𝑜 ∈ On
9 elmapg 7815 . . . . . 6 ((2𝑜 ∈ On ∧ 𝐴𝑉) → (𝑥 ∈ (2𝑜𝑚 𝐴) ↔ 𝑥:𝐴⟶2𝑜))
108, 9mpan 705 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (2𝑜𝑚 𝐴) ↔ 𝑥:𝐴⟶2𝑜))
1110anbi1d 740 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1𝑜})) ↔ (𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜}))))
12 1on 7512 . . . . . . . . . . . . 13 1𝑜 ∈ On
1312elexi 3199 . . . . . . . . . . . 12 1𝑜 ∈ V
1413sucid 5763 . . . . . . . . . . 11 1𝑜 ∈ suc 1𝑜
15 df-2o 7506 . . . . . . . . . . 11 2𝑜 = suc 1𝑜
1614, 15eleqtrri 2697 . . . . . . . . . 10 1𝑜 ∈ 2𝑜
17 0ex 4750 . . . . . . . . . . . 12 ∅ ∈ V
1817prid1 4267 . . . . . . . . . . 11 ∅ ∈ {∅, {∅}}
19 df2o2 7519 . . . . . . . . . . 11 2𝑜 = {∅, {∅}}
2018, 19eleqtrri 2697 . . . . . . . . . 10 ∅ ∈ 2𝑜
2116, 20keepel 4127 . . . . . . . . 9 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜
2221rgenw 2919 . . . . . . . 8 𝑧𝐴 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜
23 eqid 2621 . . . . . . . . 9 (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))
2423fmpt 6337 . . . . . . . 8 (∀𝑧𝐴 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜 ↔ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜)
2522, 24mpbi 220 . . . . . . 7 (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜
26 simpr 477 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))
2726feq1d 5987 . . . . . . 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 6150 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
30 elequ1 1994 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧𝑦𝑤𝑦))
3130ifbid 4080 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → if(𝑧𝑦, 1𝑜, ∅) = if(𝑤𝑦, 1𝑜, ∅))
3213, 17keepel 4127 . . . . . . . . . . . . . 14 if(𝑤𝑦, 1𝑜, ∅) ∈ V
3331, 23, 32fvmpt 6239 . . . . . . . . . . . . 13 (𝑤𝐴 → ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤) = if(𝑤𝑦, 1𝑜, ∅))
3429, 33sylan9eq 2675 . . . . . . . . . . . 12 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
3534eqeq1d 2623 . . . . . . . . . . 11 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1𝑜 ↔ if(𝑤𝑦, 1𝑜, ∅) = 1𝑜))
36 iftrue 4064 . . . . . . . . . . . 12 (𝑤𝑦 → if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
37 noel 3895 . . . . . . . . . . . . . 14 ¬ ∅ ∈ ∅
38 iffalse 4067 . . . . . . . . . . . . . . . 16 𝑤𝑦 → if(𝑤𝑦, 1𝑜, ∅) = ∅)
3938eqeq1d 2623 . . . . . . . . . . . . . . 15 𝑤𝑦 → (if(𝑤𝑦, 1𝑜, ∅) = 1𝑜 ↔ ∅ = 1𝑜))
40 0lt1o 7529 . . . . . . . . . . . . . . . 16 ∅ ∈ 1𝑜
41 eleq2 2687 . . . . . . . . . . . . . . . 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 6158 . . . . . . . . . . 11 (𝑥𝑤) ∈ V
4948elsn 4163 . . . . . . . . . 10 ((𝑥𝑤) ∈ {1𝑜} ↔ (𝑥𝑤) = 1𝑜)
5047, 49syl6bbr 278 . . . . . . . . 9 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) ∈ {1𝑜}))
5150pm5.32da 672 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → ((𝑤𝐴𝑤𝑦) ↔ (𝑤𝐴 ∧ (𝑥𝑤) ∈ {1𝑜})))
52 ssel 3577 . . . . . . . . . 10 (𝑦𝐴 → (𝑤𝑦𝑤𝐴))
5352adantr 481 . . . . . . . . 9 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤𝑦𝑤𝐴))
5453pm4.71rd 666 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤𝑦 ↔ (𝑤𝐴𝑤𝑦)))
55 ffn 6002 . . . . . . . . 9 (𝑥:𝐴⟶2𝑜𝑥 Fn 𝐴)
56 elpreima 6293 . . . . . . . . 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 2619 . . . . . 6 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → 𝑦 = (𝑥 “ {1𝑜}))
6028, 59jca 554 . . . . 5 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})))
61 simpr 477 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑦 = (𝑥 “ {1𝑜}))
62 cnvimass 5444 . . . . . . . 8 (𝑥 “ {1𝑜}) ⊆ dom 𝑥
63 fdm 6008 . . . . . . . . 9 (𝑥:𝐴⟶2𝑜 → dom 𝑥 = 𝐴)
6463adantr 481 . . . . . . . 8 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → dom 𝑥 = 𝐴)
6562, 64syl5sseq 3632 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → (𝑥 “ {1𝑜}) ⊆ 𝐴)
6661, 65eqsstrd 3618 . . . . . 6 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑦𝐴)
67 simplr 791 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → 𝑦 = (𝑥 “ {1𝑜}))
6867eleq2d 2684 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑤𝑦𝑤 ∈ (𝑥 “ {1𝑜})))
6955adantr 481 . . . . . . . . . . . . . . 15 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑥 Fn 𝐴)
70 fnbrfvb 6193 . . . . . . . . . . . . . . 15 ((𝑥 Fn 𝐴𝑤𝐴) → ((𝑥𝑤) = 1𝑜𝑤𝑥1𝑜))
7169, 70sylan 488 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1𝑜𝑤𝑥1𝑜))
72 vex 3189 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
7372eliniseg 5453 . . . . . . . . . . . . . . 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 501 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = 1𝑜)
7836adantl 482 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
7977, 78eqtr4d 2658 . . . . . . . . . 10 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
80 ffvelrn 6313 . . . . . . . . . . . . . . . . . 18 ((𝑥:𝐴⟶2𝑜𝑤𝐴) → (𝑥𝑤) ∈ 2𝑜)
8180adantlr 750 . . . . . . . . . . . . . . . . 17 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ 2𝑜)
82 df2o3 7518 . . . . . . . . . . . . . . . . 17 2𝑜 = {∅, 1𝑜}
8381, 82syl6eleq 2708 . . . . . . . . . . . . . . . 16 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ {∅, 1𝑜})
8448elpr 4169 . . . . . . . . . . . . . . . 16 ((𝑥𝑤) ∈ {∅, 1𝑜} ↔ ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1𝑜))
8583, 84sylib 208 . . . . . . . . . . . . . . 15 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1𝑜))
8685ord 392 . . . . . . . . . . . . . 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 445 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = ∅)
9038adantl 482 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → if(𝑤𝑦, 1𝑜, ∅) = ∅)
9189, 90eqtr4d 2658 . . . . . . . . . 10 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9279, 91pm2.61dan 831 . . . . . . . . 9 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9333adantl 482 . . . . . . . . 9 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9492, 93eqtr4d 2658 . . . . . . . 8 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
9594ralrimiva 2960 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
96 ffn 6002 . . . . . . . . 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 6267 . . . . . . . 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 247 . . . . . 6 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))
10166, 100jca 554 . . . . 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 4137 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
105104anbi1i 730 . . 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 6837 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 383  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3555  c0 3891  ifcif 4058  𝒫 cpw 4130  {csn 4148  {cpr 4150   class class class wbr 4613  cmpt 4673  ccnv 5073  dom cdm 5074  cima 5077  Oncon0 5682  suc csuc 5684   Fn wfn 5842  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  1𝑜c1o 7498  2𝑜c2o 7499  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1o 7505  df-2o 7506  df-map 7804
This theorem is referenced by:  pw2f1o2  37085
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