Theorem f1oen3g 6784

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6787 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 5471	. . . 4 |- ( f = F -> ( f : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
2	1	spcegv 2840	. . 3 |- ( F e. V -> ( F : A -1-1-onto-> B -> E. f f : A -1-1-onto-> B ) )
3	2	imp 124	. 2 |- ( ( F e. V /\ F : A -1-1-onto-> B ) -> E. f f : A -1-1-onto-> B )
4		bren 6777	. 2 |- ( A ~~ B <-> E. f f : A -1-1-onto-> B )
5	3, 4	sylibr 134	1 |- ( ( F e. V /\ F : A -1-1-onto-> B ) -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1503   e. wcel 2160   class class class wbr 4021   -1-1-onto->wf1o 5237   ~~ cen 6768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-en 6771

This theorem is referenced by:  f1oen2g  6785  unen  6846  phplem2  6885  sbthlemi10  6999