Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pw2f1ocnv Structured version   Visualization version   GIF version

Theorem pw2f1ocnv 43026
Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 9118, 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 3482 . . . 4 𝑥 ∈ V
32cnvex 7948 . . 3 𝑥 ∈ V
4 imaexg 7936 . . 3 (𝑥 ∈ V → (𝑥 “ {1o}) ∈ V)
53, 4mp1i 13 . 2 ((𝐴𝑉𝑥 ∈ (2om 𝐴)) → (𝑥 “ {1o}) ∈ V)
6 mptexg 7241 . . 3 (𝐴𝑉 → (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)) ∈ V)
76adantr 480 . 2 ((𝐴𝑉𝑦 ∈ 𝒫 𝐴) → (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)) ∈ V)
8 2on 8519 . . . . . 6 2o ∈ On
9 elmapg 8878 . . . . . 6 ((2o ∈ On ∧ 𝐴𝑉) → (𝑥 ∈ (2om 𝐴) ↔ 𝑥:𝐴⟶2o))
108, 9mpan 690 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (2om 𝐴) ↔ 𝑥:𝐴⟶2o))
1110anbi1d 631 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ∧ 𝑦 = (𝑥 “ {1o})) ↔ (𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o}))))
12 1oex 8515 . . . . . . . . . . . 12 1o ∈ V
1312sucid 6468 . . . . . . . . . . 11 1o ∈ suc 1o
14 df-2o 8506 . . . . . . . . . . 11 2o = suc 1o
1513, 14eleqtrri 2838 . . . . . . . . . 10 1o ∈ 2o
16 0ex 5313 . . . . . . . . . . . 12 ∅ ∈ V
1716prid1 4767 . . . . . . . . . . 11 ∅ ∈ {∅, {∅}}
18 df2o2 8514 . . . . . . . . . . 11 2o = {∅, {∅}}
1917, 18eleqtrri 2838 . . . . . . . . . 10 ∅ ∈ 2o
2015, 19ifcli 4578 . . . . . . . . 9 if(𝑧𝑦, 1o, ∅) ∈ 2o
2120rgenw 3063 . . . . . . . 8 𝑧𝐴 if(𝑧𝑦, 1o, ∅) ∈ 2o
22 eqid 2735 . . . . . . . . 9 (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)) = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))
2322fmpt 7130 . . . . . . . 8 (∀𝑧𝐴 if(𝑧𝑦, 1o, ∅) ∈ 2o ↔ (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)):𝐴⟶2o)
2421, 23mpbi 230 . . . . . . 7 (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)):𝐴⟶2o
25 simpr 484 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)))
2625feq1d 6721 . . . . . . 7 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → (𝑥:𝐴⟶2o ↔ (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)):𝐴⟶2o))
2724, 26mpbiri 258 . . . . . 6 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → 𝑥:𝐴⟶2o)
28 iftrue 4537 . . . . . . . . . . . 12 (𝑤𝑦 → if(𝑤𝑦, 1o, ∅) = 1o)
29 noel 4344 . . . . . . . . . . . . . 14 ¬ ∅ ∈ ∅
30 iffalse 4540 . . . . . . . . . . . . . . . 16 𝑤𝑦 → if(𝑤𝑦, 1o, ∅) = ∅)
3130eqeq1d 2737 . . . . . . . . . . . . . . 15 𝑤𝑦 → (if(𝑤𝑦, 1o, ∅) = 1o ↔ ∅ = 1o))
32 0lt1o 8541 . . . . . . . . . . . . . . . 16 ∅ ∈ 1o
33 eleq2 2828 . . . . . . . . . . . . . . . 16 (∅ = 1o → (∅ ∈ ∅ ↔ ∅ ∈ 1o))
3432, 33mpbiri 258 . . . . . . . . . . . . . . 15 (∅ = 1o → ∅ ∈ ∅)
3531, 34biimtrdi 253 . . . . . . . . . . . . . 14 𝑤𝑦 → (if(𝑤𝑦, 1o, ∅) = 1o → ∅ ∈ ∅))
3629, 35mtoi 199 . . . . . . . . . . . . 13 𝑤𝑦 → ¬ if(𝑤𝑦, 1o, ∅) = 1o)
3736con4i 114 . . . . . . . . . . . 12 (if(𝑤𝑦, 1o, ∅) = 1o𝑤𝑦)
3828, 37impbii 209 . . . . . . . . . . 11 (𝑤𝑦 ↔ if(𝑤𝑦, 1o, ∅) = 1o)
3925fveq1d 6909 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤))
40 elequ1 2113 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧𝑦𝑤𝑦))
4140ifbid 4554 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → if(𝑧𝑦, 1o, ∅) = if(𝑤𝑦, 1o, ∅))
4212, 16ifcli 4578 . . . . . . . . . . . . . 14 if(𝑤𝑦, 1o, ∅) ∈ V
4341, 22, 42fvmpt 7016 . . . . . . . . . . . . 13 (𝑤𝐴 → ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤) = if(𝑤𝑦, 1o, ∅))
4439, 43sylan9eq 2795 . . . . . . . . . . . 12 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1o, ∅))
4544eqeq1d 2737 . . . . . . . . . . 11 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1o ↔ if(𝑤𝑦, 1o, ∅) = 1o))
4638, 45bitr4id 290 . . . . . . . . . 10 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) = 1o))
47 fvex 6920 . . . . . . . . . . 11 (𝑥𝑤) ∈ V
4847elsn 4646 . . . . . . . . . 10 ((𝑥𝑤) ∈ {1o} ↔ (𝑥𝑤) = 1o)
4946, 48bitr4di 289 . . . . . . . . 9 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) ∈ {1o}))
5049pm5.32da 579 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → ((𝑤𝐴𝑤𝑦) ↔ (𝑤𝐴 ∧ (𝑥𝑤) ∈ {1o})))
51 ssel 3989 . . . . . . . . . 10 (𝑦𝐴 → (𝑤𝑦𝑤𝐴))
5251adantr 480 . . . . . . . . 9 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → (𝑤𝑦𝑤𝐴))
5352pm4.71rd 562 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → (𝑤𝑦 ↔ (𝑤𝐴𝑤𝑦)))
54 ffn 6737 . . . . . . . . 9 (𝑥:𝐴⟶2o𝑥 Fn 𝐴)
55 elpreima 7078 . . . . . . . . 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 311 . . . . . . 7 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → (𝑤𝑦𝑤 ∈ (𝑥 “ {1o})))
5857eqrdv 2733 . . . . . 6 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → 𝑦 = (𝑥 “ {1o}))
5927, 58jca 511 . . . . 5 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → (𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})))
60 simpr 484 . . . . . . 7 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → 𝑦 = (𝑥 “ {1o}))
61 cnvimass 6102 . . . . . . . 8 (𝑥 “ {1o}) ⊆ dom 𝑥
62 fdm 6746 . . . . . . . . 9 (𝑥:𝐴⟶2o → dom 𝑥 = 𝐴)
6362adantr 480 . . . . . . . 8 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → dom 𝑥 = 𝐴)
6461, 63sseqtrid 4048 . . . . . . 7 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → (𝑥 “ {1o}) ⊆ 𝐴)
6560, 64eqsstrd 4034 . . . . . 6 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → 𝑦𝐴)
66 simplr 769 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → 𝑦 = (𝑥 “ {1o}))
6766eleq2d 2825 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑤𝑦𝑤 ∈ (𝑥 “ {1o})))
6854adantr 480 . . . . . . . . . . . . . . 15 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → 𝑥 Fn 𝐴)
69 fnbrfvb 6960 . . . . . . . . . . . . . . 15 ((𝑥 Fn 𝐴𝑤𝐴) → ((𝑥𝑤) = 1o𝑤𝑥1o))
7068, 69sylan 580 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1o𝑤𝑥1o))
71 1on 8517 . . . . . . . . . . . . . . 15 1o ∈ On
72 vex 3482 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
7372eliniseg 6115 . . . . . . . . . . . . . . 15 (1o ∈ On → (𝑤 ∈ (𝑥 “ {1o}) ↔ 𝑤𝑥1o))
7471, 73ax-mp 5 . . . . . . . . . . . . . 14 (𝑤 ∈ (𝑥 “ {1o}) ↔ 𝑤𝑥1o)
7570, 74bitr4di 289 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1o𝑤 ∈ (𝑥 “ {1o})))
7667, 75bitr4d 282 . . . . . . . . . . . 12 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) = 1o))
7776biimpa 476 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = 1o)
7828adantl 481 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → if(𝑤𝑦, 1o, ∅) = 1o)
7977, 78eqtr4d 2778 . . . . . . . . . 10 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1o, ∅))
80 ffvelcdm 7101 . . . . . . . . . . . . . . . . . 18 ((𝑥:𝐴⟶2o𝑤𝐴) → (𝑥𝑤) ∈ 2o)
8180adantlr 715 . . . . . . . . . . . . . . . . 17 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ 2o)
82 df2o3 8513 . . . . . . . . . . . . . . . . 17 2o = {∅, 1o}
8381, 82eleqtrdi 2849 . . . . . . . . . . . . . . . 16 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ {∅, 1o})
8447elpr 4655 . . . . . . . . . . . . . . . 16 ((𝑥𝑤) ∈ {∅, 1o} ↔ ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1o))
8583, 84sylib 218 . . . . . . . . . . . . . . 15 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1o))
8685ord 864 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (¬ (𝑥𝑤) = ∅ → (𝑥𝑤) = 1o))
8786, 76sylibrd 259 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (¬ (𝑥𝑤) = ∅ → 𝑤𝑦))
8887con1d 145 . . . . . . . . . . . 12 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (¬ 𝑤𝑦 → (𝑥𝑤) = ∅))
8988imp 406 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = ∅)
9030adantl 481 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → if(𝑤𝑦, 1o, ∅) = ∅)
9189, 90eqtr4d 2778 . . . . . . . . . 10 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1o, ∅))
9279, 91pm2.61dan 813 . . . . . . . . 9 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1o, ∅))
9343adantl 481 . . . . . . . . 9 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤) = if(𝑤𝑦, 1o, ∅))
9492, 93eqtr4d 2778 . . . . . . . 8 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤))
9594ralrimiva 3144 . . . . . . 7 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤))
96 ffn 6737 . . . . . . . . 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 7051 . . . . . . . 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 586 . . . . . . 7 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → (𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)) ↔ ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤)))
10095, 99mpbird 257 . . . . . 6 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)))
10165, 100jca 511 . . . . 5 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))))
10259, 101impbii 209 . . . 4 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ↔ (𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})))
10311, 102bitr4di 289 . . 3 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ∧ 𝑦 = (𝑥 “ {1o})) ↔ (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)))))
104 velpw 4610 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
105104anbi1i 624 . . 3 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ↔ (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))))
106103, 105bitr4di 289 . 2 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ∧ 𝑦 = (𝑥 “ {1o})) ↔ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)))))
1071, 5, 7, 106f1ocnvd 7684 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 206  wa 395  wo 847   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963  c0 4339  ifcif 4531  𝒫 cpw 4605  {csn 4631  {cpr 4633   class class class wbr 5148  cmpt 5231  ccnv 5688  dom cdm 5689  cima 5692  Oncon0 6386  suc csuc 6388   Fn wfn 6558  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  1oc1o 8498  2oc2o 8499  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8505  df-2o 8506  df-map 8867
This theorem is referenced by:  pw2f1o2  43027
  Copyright terms: Public domain W3C validator