Theorem f1oen3g 6720

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6723 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 5421	. . . 4 |- ( f = F -> ( f : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
2	1	spcegv 2814	. . 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 6713	. 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 1480   e. wcel 2136   class class class wbr 3982   -1-1-onto->wf1o 5187   ~~ cen 6704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-en 6707

This theorem is referenced by:  f1oen2g  6721  unen  6782  phplem2  6819  sbthlemi10  6931