Theorem f1oen3g 6527

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6530 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
f1oen3g	|- ( ( F e. V /\ F : A -1-1-onto-> B ) -> A ~~ B )

Proof of Theorem f1oen3g

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oeq1 5259	. . . 4 |- ( f = F -> ( f : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
2	1	spcegv 2710	. . 3 |- ( F e. V -> ( F : A -1-1-onto-> B -> E. f f : A -1-1-onto-> B ) )
3	2	imp 123	. 2 |- ( ( F e. V /\ F : A -1-1-onto-> B ) -> E. f f : A -1-1-onto-> B )
4		bren 6520	. 2 |- ( A ~~ B <-> E. f f : A -1-1-onto-> B )
5	3, 4	sylibr 133	1 |- ( ( F e. V /\ F : A -1-1-onto-> B ) -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1427   e. wcel 1439   class class class wbr 3853   -1-1-onto->wf1o 5029   ~~ cen 6511

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-en 6514

This theorem is referenced by:  f1oen2g  6528  unen  6589  phplem2  6625  sbthlemi10  6731