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Theorem pw2f1ocnv 38281
Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8274, 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 3353 . . . 4 𝑥 ∈ V
32cnvex 7311 . . 3 𝑥 ∈ V
4 imaexg 7301 . . 3 (𝑥 ∈ V → (𝑥 “ {1𝑜}) ∈ V)
53, 4mp1i 13 . 2 ((𝐴𝑉𝑥 ∈ (2𝑜𝑚 𝐴)) → (𝑥 “ {1𝑜}) ∈ V)
6 mptexg 6677 . . 3 (𝐴𝑉 → (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ∈ V)
76adantr 472 . 2 ((𝐴𝑉𝑦 ∈ 𝒫 𝐴) → (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ∈ V)
8 2on 7773 . . . . . 6 2𝑜 ∈ On
9 elmapg 8073 . . . . . 6 ((2𝑜 ∈ On ∧ 𝐴𝑉) → (𝑥 ∈ (2𝑜𝑚 𝐴) ↔ 𝑥:𝐴⟶2𝑜))
108, 9mpan 681 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (2𝑜𝑚 𝐴) ↔ 𝑥:𝐴⟶2𝑜))
1110anbi1d 623 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1𝑜})) ↔ (𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜}))))
12 1oex 7772 . . . . . . . . . . . 12 1𝑜 ∈ V
1312sucid 5987 . . . . . . . . . . 11 1𝑜 ∈ suc 1𝑜
14 df-2o 7765 . . . . . . . . . . 11 2𝑜 = suc 1𝑜
1513, 14eleqtrri 2843 . . . . . . . . . 10 1𝑜 ∈ 2𝑜
16 0ex 4950 . . . . . . . . . . . 12 ∅ ∈ V
1716prid1 4452 . . . . . . . . . . 11 ∅ ∈ {∅, {∅}}
18 df2o2 7779 . . . . . . . . . . 11 2𝑜 = {∅, {∅}}
1917, 18eleqtrri 2843 . . . . . . . . . 10 ∅ ∈ 2𝑜
2015, 19ifcli 4289 . . . . . . . . 9 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜
2120rgenw 3071 . . . . . . . 8 𝑧𝐴 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜
22 eqid 2765 . . . . . . . . 9 (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))
2322fmpt 6570 . . . . . . . 8 (∀𝑧𝐴 if(𝑧𝑦, 1𝑜, ∅) ∈ 2𝑜 ↔ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜)
2421, 23mpbi 221 . . . . . . 7 (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜
25 simpr 477 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))
2625feq1d 6208 . . . . . . 7 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑥:𝐴⟶2𝑜 ↔ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜))
2724, 26mpbiri 249 . . . . . 6 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → 𝑥:𝐴⟶2𝑜)
2825fveq1d 6377 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
29 elequ1 2162 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧𝑦𝑤𝑦))
3029ifbid 4265 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → if(𝑧𝑦, 1𝑜, ∅) = if(𝑤𝑦, 1𝑜, ∅))
3112, 16ifcli 4289 . . . . . . . . . . . . . 14 if(𝑤𝑦, 1𝑜, ∅) ∈ V
3230, 22, 31fvmpt 6471 . . . . . . . . . . . . 13 (𝑤𝐴 → ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤) = if(𝑤𝑦, 1𝑜, ∅))
3328, 32sylan9eq 2819 . . . . . . . . . . . 12 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
3433eqeq1d 2767 . . . . . . . . . . 11 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1𝑜 ↔ if(𝑤𝑦, 1𝑜, ∅) = 1𝑜))
35 iftrue 4249 . . . . . . . . . . . 12 (𝑤𝑦 → if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
36 noel 4083 . . . . . . . . . . . . . 14 ¬ ∅ ∈ ∅
37 iffalse 4252 . . . . . . . . . . . . . . . 16 𝑤𝑦 → if(𝑤𝑦, 1𝑜, ∅) = ∅)
3837eqeq1d 2767 . . . . . . . . . . . . . . 15 𝑤𝑦 → (if(𝑤𝑦, 1𝑜, ∅) = 1𝑜 ↔ ∅ = 1𝑜))
39 0lt1o 7789 . . . . . . . . . . . . . . . 16 ∅ ∈ 1𝑜
40 eleq2 2833 . . . . . . . . . . . . . . . 16 (∅ = 1𝑜 → (∅ ∈ ∅ ↔ ∅ ∈ 1𝑜))
4139, 40mpbiri 249 . . . . . . . . . . . . . . 15 (∅ = 1𝑜 → ∅ ∈ ∅)
4238, 41syl6bi 244 . . . . . . . . . . . . . 14 𝑤𝑦 → (if(𝑤𝑦, 1𝑜, ∅) = 1𝑜 → ∅ ∈ ∅))
4336, 42mtoi 190 . . . . . . . . . . . . 13 𝑤𝑦 → ¬ if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
4443con4i 114 . . . . . . . . . . . 12 (if(𝑤𝑦, 1𝑜, ∅) = 1𝑜𝑤𝑦)
4535, 44impbii 200 . . . . . . . . . . 11 (𝑤𝑦 ↔ if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
4634, 45syl6rbbr 281 . . . . . . . . . 10 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) = 1𝑜))
47 fvex 6388 . . . . . . . . . . 11 (𝑥𝑤) ∈ V
4847elsn 4349 . . . . . . . . . 10 ((𝑥𝑤) ∈ {1𝑜} ↔ (𝑥𝑤) = 1𝑜)
4946, 48syl6bbr 280 . . . . . . . . 9 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) ∈ {1𝑜}))
5049pm5.32da 574 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → ((𝑤𝐴𝑤𝑦) ↔ (𝑤𝐴 ∧ (𝑥𝑤) ∈ {1𝑜})))
51 ssel 3755 . . . . . . . . . 10 (𝑦𝐴 → (𝑤𝑦𝑤𝐴))
5251adantr 472 . . . . . . . . 9 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤𝑦𝑤𝐴))
5352pm4.71rd 558 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤𝑦 ↔ (𝑤𝐴𝑤𝑦)))
54 ffn 6223 . . . . . . . . 9 (𝑥:𝐴⟶2𝑜𝑥 Fn 𝐴)
55 elpreima 6527 . . . . . . . . 9 (𝑥 Fn 𝐴 → (𝑤 ∈ (𝑥 “ {1𝑜}) ↔ (𝑤𝐴 ∧ (𝑥𝑤) ∈ {1𝑜})))
5627, 54, 553syl 18 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤 ∈ (𝑥 “ {1𝑜}) ↔ (𝑤𝐴 ∧ (𝑥𝑤) ∈ {1𝑜})))
5750, 53, 563bitr4d 302 . . . . . . 7 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑤𝑦𝑤 ∈ (𝑥 “ {1𝑜})))
5857eqrdv 2763 . . . . . 6 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → 𝑦 = (𝑥 “ {1𝑜}))
5927, 58jca 507 . . . . 5 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) → (𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})))
60 simpr 477 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑦 = (𝑥 “ {1𝑜}))
61 cnvimass 5667 . . . . . . . 8 (𝑥 “ {1𝑜}) ⊆ dom 𝑥
62 fdm 6231 . . . . . . . . 9 (𝑥:𝐴⟶2𝑜 → dom 𝑥 = 𝐴)
6362adantr 472 . . . . . . . 8 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → dom 𝑥 = 𝐴)
6461, 63syl5sseq 3813 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → (𝑥 “ {1𝑜}) ⊆ 𝐴)
6560, 64eqsstrd 3799 . . . . . 6 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑦𝐴)
66 simplr 785 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → 𝑦 = (𝑥 “ {1𝑜}))
6766eleq2d 2830 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑤𝑦𝑤 ∈ (𝑥 “ {1𝑜})))
6854adantr 472 . . . . . . . . . . . . . . 15 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑥 Fn 𝐴)
69 fnbrfvb 6424 . . . . . . . . . . . . . . 15 ((𝑥 Fn 𝐴𝑤𝐴) → ((𝑥𝑤) = 1𝑜𝑤𝑥1𝑜))
7068, 69sylan 575 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1𝑜𝑤𝑥1𝑜))
71 1on 7771 . . . . . . . . . . . . . . 15 1𝑜 ∈ On
72 vex 3353 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
7372eliniseg 5676 . . . . . . . . . . . . . . 15 (1𝑜 ∈ On → (𝑤 ∈ (𝑥 “ {1𝑜}) ↔ 𝑤𝑥1𝑜))
7471, 73ax-mp 5 . . . . . . . . . . . . . 14 (𝑤 ∈ (𝑥 “ {1𝑜}) ↔ 𝑤𝑥1𝑜)
7570, 74syl6bbr 280 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1𝑜𝑤 ∈ (𝑥 “ {1𝑜})))
7667, 75bitr4d 273 . . . . . . . . . . . 12 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) = 1𝑜))
7776biimpa 468 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = 1𝑜)
7835adantl 473 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → if(𝑤𝑦, 1𝑜, ∅) = 1𝑜)
7977, 78eqtr4d 2802 . . . . . . . . . 10 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
80 ffvelrn 6547 . . . . . . . . . . . . . . . . . 18 ((𝑥:𝐴⟶2𝑜𝑤𝐴) → (𝑥𝑤) ∈ 2𝑜)
8180adantlr 706 . . . . . . . . . . . . . . . . 17 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ 2𝑜)
82 df2o3 7778 . . . . . . . . . . . . . . . . 17 2𝑜 = {∅, 1𝑜}
8381, 82syl6eleq 2854 . . . . . . . . . . . . . . . 16 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ {∅, 1𝑜})
8447elpr 4357 . . . . . . . . . . . . . . . 16 ((𝑥𝑤) ∈ {∅, 1𝑜} ↔ ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1𝑜))
8583, 84sylib 209 . . . . . . . . . . . . . . 15 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1𝑜))
8685ord 890 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (¬ (𝑥𝑤) = ∅ → (𝑥𝑤) = 1𝑜))
8786, 76sylibrd 250 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (¬ (𝑥𝑤) = ∅ → 𝑤𝑦))
8887con1d 141 . . . . . . . . . . . 12 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (¬ 𝑤𝑦 → (𝑥𝑤) = ∅))
8988imp 395 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = ∅)
9037adantl 473 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → if(𝑤𝑦, 1𝑜, ∅) = ∅)
9189, 90eqtr4d 2802 . . . . . . . . . 10 ((((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9279, 91pm2.61dan 847 . . . . . . . . 9 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9332adantl 473 . . . . . . . . 9 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤) = if(𝑤𝑦, 1𝑜, ∅))
9492, 93eqtr4d 2802 . . . . . . . 8 (((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) ∧ 𝑤𝐴) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
9594ralrimiva 3113 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤))
96 ffn 6223 . . . . . . . . 9 ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)):𝐴⟶2𝑜 → (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) Fn 𝐴)
9724, 96ax-mp 5 . . . . . . . 8 (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) Fn 𝐴
98 eqfnfv 6501 . . . . . . . 8 ((𝑥 Fn 𝐴 ∧ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) Fn 𝐴) → (𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ↔ ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤)))
9968, 97, 98sylancl 580 . . . . . . 7 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → (𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)) ↔ ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))‘𝑤)))
10095, 99mpbird 248 . . . . . 6 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))
10165, 100jca 507 . . . . 5 ((𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})) → (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))))
10259, 101impbii 200 . . . 4 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ↔ (𝑥:𝐴⟶2𝑜𝑦 = (𝑥 “ {1𝑜})))
10311, 102syl6bbr 280 . . 3 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1𝑜})) ↔ (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))))
104 selpw 4322 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
105104anbi1i 617 . . 3 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))) ↔ (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅))))
106103, 105syl6bbr 280 . 2 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1𝑜})) ↔ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))))
1071, 5, 7, 106f1ocnvd 7082 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 197  wa 384  wo 873   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  wss 3732  c0 4079  ifcif 4243  𝒫 cpw 4315  {csn 4334  {cpr 4336   class class class wbr 4809  cmpt 4888  ccnv 5276  dom cdm 5277  cima 5280  Oncon0 5908  suc csuc 5910   Fn wfn 6063  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  1𝑜c1o 7757  2𝑜c2o 7758  𝑚 cmap 8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1o 7764  df-2o 7765  df-map 8062
This theorem is referenced by:  pw2f1o2  38282
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