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 39027
 Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8420, 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 3419 . . . 4 𝑥 ∈ V
32cnvex 7445 . . 3 𝑥 ∈ V
4 imaexg 7435 . . 3 (𝑥 ∈ V → (𝑥 “ {1o}) ∈ V)
53, 4mp1i 13 . 2 ((𝐴𝑉𝑥 ∈ (2o𝑚 𝐴)) → (𝑥 “ {1o}) ∈ V)
6 mptexg 6810 . . 3 (𝐴𝑉 → (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)) ∈ V)
76adantr 473 . 2 ((𝐴𝑉𝑦 ∈ 𝒫 𝐴) → (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)) ∈ V)
8 2on 7914 . . . . . 6 2o ∈ On
9 elmapg 8219 . . . . . 6 ((2o ∈ On ∧ 𝐴𝑉) → (𝑥 ∈ (2o𝑚 𝐴) ↔ 𝑥:𝐴⟶2o))
108, 9mpan 677 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (2o𝑚 𝐴) ↔ 𝑥:𝐴⟶2o))
1110anbi1d 620 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2o𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1o})) ↔ (𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o}))))
12 1oex 7913 . . . . . . . . . . . 12 1o ∈ V
1312sucid 6108 . . . . . . . . . . 11 1o ∈ suc 1o
14 df-2o 7906 . . . . . . . . . . 11 2o = suc 1o
1513, 14eleqtrri 2866 . . . . . . . . . 10 1o ∈ 2o
16 0ex 5068 . . . . . . . . . . . 12 ∅ ∈ V
1716prid1 4572 . . . . . . . . . . 11 ∅ ∈ {∅, {∅}}
18 df2o2 7920 . . . . . . . . . . 11 2o = {∅, {∅}}
1917, 18eleqtrri 2866 . . . . . . . . . 10 ∅ ∈ 2o
2015, 19ifcli 4396 . . . . . . . . 9 if(𝑧𝑦, 1o, ∅) ∈ 2o
2120rgenw 3101 . . . . . . . 8 𝑧𝐴 if(𝑧𝑦, 1o, ∅) ∈ 2o
22 eqid 2779 . . . . . . . . 9 (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)) = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))
2322fmpt 6697 . . . . . . . 8 (∀𝑧𝐴 if(𝑧𝑦, 1o, ∅) ∈ 2o ↔ (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)):𝐴⟶2o)
2421, 23mpbi 222 . . . . . . 7 (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)):𝐴⟶2o
25 simpr 477 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)))
2625feq1d 6329 . . . . . . 7 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → (𝑥:𝐴⟶2o ↔ (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)):𝐴⟶2o))
2724, 26mpbiri 250 . . . . . 6 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → 𝑥:𝐴⟶2o)
2825fveq1d 6501 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤))
29 elequ1 2057 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧𝑦𝑤𝑦))
3029ifbid 4372 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → if(𝑧𝑦, 1o, ∅) = if(𝑤𝑦, 1o, ∅))
3112, 16ifcli 4396 . . . . . . . . . . . . . 14 if(𝑤𝑦, 1o, ∅) ∈ V
3230, 22, 31fvmpt 6595 . . . . . . . . . . . . 13 (𝑤𝐴 → ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤) = if(𝑤𝑦, 1o, ∅))
3328, 32sylan9eq 2835 . . . . . . . . . . . 12 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1o, ∅))
3433eqeq1d 2781 . . . . . . . . . . 11 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1o ↔ if(𝑤𝑦, 1o, ∅) = 1o))
35 iftrue 4356 . . . . . . . . . . . 12 (𝑤𝑦 → if(𝑤𝑦, 1o, ∅) = 1o)
36 noel 4184 . . . . . . . . . . . . . 14 ¬ ∅ ∈ ∅
37 iffalse 4359 . . . . . . . . . . . . . . . 16 𝑤𝑦 → if(𝑤𝑦, 1o, ∅) = ∅)
3837eqeq1d 2781 . . . . . . . . . . . . . . 15 𝑤𝑦 → (if(𝑤𝑦, 1o, ∅) = 1o ↔ ∅ = 1o))
39 0lt1o 7931 . . . . . . . . . . . . . . . 16 ∅ ∈ 1o
40 eleq2 2855 . . . . . . . . . . . . . . . 16 (∅ = 1o → (∅ ∈ ∅ ↔ ∅ ∈ 1o))
4139, 40mpbiri 250 . . . . . . . . . . . . . . 15 (∅ = 1o → ∅ ∈ ∅)
4238, 41syl6bi 245 . . . . . . . . . . . . . 14 𝑤𝑦 → (if(𝑤𝑦, 1o, ∅) = 1o → ∅ ∈ ∅))
4336, 42mtoi 191 . . . . . . . . . . . . 13 𝑤𝑦 → ¬ if(𝑤𝑦, 1o, ∅) = 1o)
4443con4i 114 . . . . . . . . . . . 12 (if(𝑤𝑦, 1o, ∅) = 1o𝑤𝑦)
4535, 44impbii 201 . . . . . . . . . . 11 (𝑤𝑦 ↔ if(𝑤𝑦, 1o, ∅) = 1o)
4634, 45syl6rbbr 282 . . . . . . . . . 10 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) = 1o))
47 fvex 6512 . . . . . . . . . . 11 (𝑥𝑤) ∈ V
4847elsn 4456 . . . . . . . . . 10 ((𝑥𝑤) ∈ {1o} ↔ (𝑥𝑤) = 1o)
4946, 48syl6bbr 281 . . . . . . . . 9 (((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) ∈ {1o}))
5049pm5.32da 571 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → ((𝑤𝐴𝑤𝑦) ↔ (𝑤𝐴 ∧ (𝑥𝑤) ∈ {1o})))
51 ssel 3853 . . . . . . . . . 10 (𝑦𝐴 → (𝑤𝑦𝑤𝐴))
5251adantr 473 . . . . . . . . 9 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → (𝑤𝑦𝑤𝐴))
5352pm4.71rd 555 . . . . . . . 8 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → (𝑤𝑦 ↔ (𝑤𝐴𝑤𝑦)))
54 ffn 6344 . . . . . . . . 9 (𝑥:𝐴⟶2o𝑥 Fn 𝐴)
55 elpreima 6653 . . . . . . . . 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 303 . . . . . . 7 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → (𝑤𝑦𝑤 ∈ (𝑥 “ {1o})))
5857eqrdv 2777 . . . . . 6 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) → 𝑦 = (𝑥 “ {1o}))
5927, 58jca 504 . . . . 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 5789 . . . . . . . 8 (𝑥 “ {1o}) ⊆ dom 𝑥
62 fdm 6352 . . . . . . . . 9 (𝑥:𝐴⟶2o → dom 𝑥 = 𝐴)
6362adantr 473 . . . . . . . 8 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → dom 𝑥 = 𝐴)
6461, 63syl5sseq 3910 . . . . . . 7 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → (𝑥 “ {1o}) ⊆ 𝐴)
6560, 64eqsstrd 3896 . . . . . 6 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → 𝑦𝐴)
66 simplr 756 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → 𝑦 = (𝑥 “ {1o}))
6766eleq2d 2852 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑤𝑦𝑤 ∈ (𝑥 “ {1o})))
6854adantr 473 . . . . . . . . . . . . . . 15 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → 𝑥 Fn 𝐴)
69 fnbrfvb 6548 . . . . . . . . . . . . . . 15 ((𝑥 Fn 𝐴𝑤𝐴) → ((𝑥𝑤) = 1o𝑤𝑥1o))
7068, 69sylan 572 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1o𝑤𝑥1o))
71 1on 7912 . . . . . . . . . . . . . . 15 1o ∈ On
72 vex 3419 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
7372eliniseg 5798 . . . . . . . . . . . . . . 15 (1o ∈ On → (𝑤 ∈ (𝑥 “ {1o}) ↔ 𝑤𝑥1o))
7471, 73ax-mp 5 . . . . . . . . . . . . . 14 (𝑤 ∈ (𝑥 “ {1o}) ↔ 𝑤𝑥1o)
7570, 74syl6bbr 281 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → ((𝑥𝑤) = 1o𝑤 ∈ (𝑥 “ {1o})))
7667, 75bitr4d 274 . . . . . . . . . . . 12 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑤𝑦 ↔ (𝑥𝑤) = 1o))
7776biimpa 469 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = 1o)
7835adantl 474 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → if(𝑤𝑦, 1o, ∅) = 1o)
7977, 78eqtr4d 2818 . . . . . . . . . 10 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1o, ∅))
80 ffvelrn 6674 . . . . . . . . . . . . . . . . . 18 ((𝑥:𝐴⟶2o𝑤𝐴) → (𝑥𝑤) ∈ 2o)
8180adantlr 702 . . . . . . . . . . . . . . . . 17 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ 2o)
82 df2o3 7919 . . . . . . . . . . . . . . . . 17 2o = {∅, 1o}
8381, 82syl6eleq 2877 . . . . . . . . . . . . . . . 16 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ {∅, 1o})
8447elpr 4464 . . . . . . . . . . . . . . . 16 ((𝑥𝑤) ∈ {∅, 1o} ↔ ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1o))
8583, 84sylib 210 . . . . . . . . . . . . . . 15 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → ((𝑥𝑤) = ∅ ∨ (𝑥𝑤) = 1o))
8685ord 850 . . . . . . . . . . . . . 14 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (¬ (𝑥𝑤) = ∅ → (𝑥𝑤) = 1o))
8786, 76sylibrd 251 . . . . . . . . . . . . 13 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (¬ (𝑥𝑤) = ∅ → 𝑤𝑦))
8887con1d 142 . . . . . . . . . . . 12 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (¬ 𝑤𝑦 → (𝑥𝑤) = ∅))
8988imp 398 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = ∅)
9037adantl 474 . . . . . . . . . . 11 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → if(𝑤𝑦, 1o, ∅) = ∅)
9189, 90eqtr4d 2818 . . . . . . . . . 10 ((((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) ∧ ¬ 𝑤𝑦) → (𝑥𝑤) = if(𝑤𝑦, 1o, ∅))
9279, 91pm2.61dan 800 . . . . . . . . 9 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑥𝑤) = if(𝑤𝑦, 1o, ∅))
9332adantl 474 . . . . . . . . 9 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤) = if(𝑤𝑦, 1o, ∅))
9492, 93eqtr4d 2818 . . . . . . . 8 (((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) ∧ 𝑤𝐴) → (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤))
9594ralrimiva 3133 . . . . . . 7 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤))
96 ffn 6344 . . . . . . . . 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 6627 . . . . . . . 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 577 . . . . . . 7 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → (𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)) ↔ ∀𝑤𝐴 (𝑥𝑤) = ((𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))‘𝑤)))
10095, 99mpbird 249 . . . . . 6 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → 𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)))
10165, 100jca 504 . . . . 5 ((𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})) → (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))))
10259, 101impbii 201 . . . 4 ((𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ↔ (𝑥:𝐴⟶2o𝑦 = (𝑥 “ {1o})))
10311, 102syl6bbr 281 . . 3 (𝐴𝑉 → ((𝑥 ∈ (2o𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1o})) ↔ (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)))))
104 selpw 4429 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
105104anbi1i 614 . . 3 ((𝑦 ∈ 𝒫 𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))) ↔ (𝑦𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅))))
106103, 105syl6bbr 281 . 2 (𝐴𝑉 → ((𝑥 ∈ (2o𝑚 𝐴) ∧ 𝑦 = (𝑥 “ {1o})) ↔ (𝑦 ∈ 𝒫 𝐴𝑥 = (𝑧𝐴 ↦ if(𝑧𝑦, 1o, ∅)))))
1071, 5, 7, 106f1ocnvd 7214 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 198   ∧ wa 387   ∨ wo 833   = wceq 1507   ∈ wcel 2050  ∀wral 3089  Vcvv 3416   ⊆ wss 3830  ∅c0 4179  ifcif 4350  𝒫 cpw 4422  {csn 4441  {cpr 4443   class class class wbr 4929   ↦ cmpt 5008  ◡ccnv 5406  dom cdm 5407   “ cima 5410  Oncon0 6029  suc csuc 6031   Fn wfn 6183  ⟶wf 6184  –1-1-onto→wf1o 6187  ‘cfv 6188  (class class class)co 6976  1oc1o 7898  2oc2o 7899   ↑𝑚 cmap 8206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-ord 6032  df-on 6033  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1o 7905  df-2o 7906  df-map 8208 This theorem is referenced by:  pw2f1o2  39028
 Copyright terms: Public domain W3C validator