Theorem f1oexbi 7116

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	|- ( E. f f : A -1-1-onto-> B <-> E. g g : B -1-1-onto-> A )

Distinct variable groups:   A, f, g   B, f, g

Proof of Theorem f1oexbi

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- f e. _V
2	1	cnvex 7113	. . . 4 |- `' f e. _V
3		f1ocnv 6149	. . . 4 |- ( f : A -1-1-onto-> B -> `' f : B -1-1-onto-> A )
4		f1oeq1 6127	. . . . 5 |- ( g = `' f -> ( g : B -1-1-onto-> A <-> `' f : B -1-1-onto-> A ) )
5	4	spcegv 3294	. . . 4 |- ( `' f e. _V -> ( `' f : B -1-1-onto-> A -> E. g g : B -1-1-onto-> A ) )
6	2, 3, 5	mpsyl 68	. . 3 |- ( f : A -1-1-onto-> B -> E. g g : B -1-1-onto-> A )
7	6	exlimiv 1858	. 2 |- ( E. f f : A -1-1-onto-> B -> E. g g : B -1-1-onto-> A )
8		vex 3203	. . . . 5 |- g e. _V
9	8	cnvex 7113	. . . 4 |- `' g e. _V
10		f1ocnv 6149	. . . 4 |- ( g : B -1-1-onto-> A -> `' g : A -1-1-onto-> B )
11		f1oeq1 6127	. . . . 5 |- ( f = `' g -> ( f : A -1-1-onto-> B <-> `' g : A -1-1-onto-> B ) )
12	11	spcegv 3294	. . . 4 |- ( `' g e. _V -> ( `' g : A -1-1-onto-> B -> E. f f : A -1-1-onto-> B ) )
13	9, 10, 12	mpsyl 68	. . 3 |- ( g : B -1-1-onto-> A -> E. f f : A -1-1-onto-> B )
14	13	exlimiv 1858	. 2 |- ( E. g g : B -1-1-onto-> A -> E. f f : A -1-1-onto-> B )
15	7, 14	impbii 199	1 |- ( E. f f : A -1-1-onto-> B <-> E. g g : B -1-1-onto-> A )

Syntax hints:   <-> wb 196   E.wex 1704   e. wcel 1990   _Vcvv 3200   `'ccnv 5113   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  rusgrnumwlkg  26872  f1ocnt  29559