Theorem f1oen3g 7044

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7047 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 5630	. . . 4 |- ( f = F -> ( f : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
2	1	spcegv 2913	. . 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 7034	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   E.wex 1545   e. wcel 2209   class class class wbr 4131   -1-1-onto->wf1o 5379   ~~ cen 7024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-en 7027

This theorem is used by:  f1oen2g  7045  unen  7109  phplem2  7158  sbthlemi10  7288  hashf1lem2  11308