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Theorem pw2f1o 7578
 Description: The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.)
Hypotheses
Ref Expression
pw2f1o.1
pw2f1o.2
pw2f1o.3
pw2f1o.4
pw2f1o.5
Assertion
Ref Expression
pw2f1o
Distinct variable groups:   ,,   ,,   ,,   ,
Allowed substitution hints:   ()   (,)   (,)   (,)

Proof of Theorem pw2f1o
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pw2f1o.5 . 2
2 eqid 2400 . . . 4
3 pw2f1o.1 . . . . . 6
4 pw2f1o.2 . . . . . 6
5 pw2f1o.3 . . . . . 6
6 pw2f1o.4 . . . . . 6
73, 4, 5, 6pw2f1olem 7577 . . . . 5
87biimpa 482 . . . 4
92, 8mpanr2 682 . . 3
109simpld 457 . 2
11 vex 3059 . . . 4
1211cnvex 6683 . . 3
13 imaexg 6673 . . 3
1412, 13mp1i 13 . 2
153, 4, 5, 6pw2f1olem 7577 . 2
161, 10, 14, 15f1od 6460 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   wceq 1403   wcel 1840   wne 2596  cvv 3056  cif 3882  cpw 3952  csn 3969  cpr 3971   cmpt 4450  ccnv 4939  cima 4943  wf1o 5522  (class class class)co 6232   cmap 7375 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-map 7377 This theorem is referenced by:  pw2eng  7579  indf1o  28352
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