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Theorem f1oexbi 6750
Description: There is a one-to-one onto function from a set to a second set iff there is a one-to-one onto function from the second set to the first set. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
Assertion
Ref Expression
f1oexbi
Distinct variable groups:   , ,   , ,

Proof of Theorem f1oexbi
StepHypRef Expression
1 vex 3112 . . . . 5
21cnvex 6747 . . . 4
3 f1ocnv 5833 . . . 4
4 f1oeq1 5812 . . . . 5
54spcegv 3195 . . . 4
62, 3, 5mpsyl 63 . . 3
76exlimiv 1722 . 2
8 vex 3112 . . . . 5
98cnvex 6747 . . . 4
10 f1ocnv 5833 . . . 4
11 f1oeq1 5812 . . . . 5
1211spcegv 3195 . . . 4
139, 10, 12mpsyl 63 . . 3
1413exlimiv 1722 . 2
157, 14impbii 188 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  E.wex 1612  e.wcel 1818   cvv 3109  `'ccnv 5003  -1-1-onto->wf1o 5592
This theorem is referenced by:  rusgranumwwlkg  24959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600
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