Theorem f1oen2g 7041

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7043 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
f1oen2g	|- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> A ~~ B )

Proof of Theorem f1oen2g

Step	Hyp	Ref	Expression
1		f1of 5639	. . . 4 |- ( F : A -1-1-onto-> B -> F : A --> B )
2		fex2 5556	. . . 4 |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V )
3	1, 2	syl3an1 1311	. . 3 |- ( ( F : A -1-1-onto-> B /\ A e. V /\ B e. W ) -> F e. _V )
4	3	3coml 1241	. 2 |- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> F e. _V )
5		simp3 1030	. 2 |- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> F : A -1-1-onto-> B )
6		f1oen3g 7040	. 2 |- ( ( F e. _V /\ F : A -1-1-onto-> B ) -> A ~~ B )
7	4, 5, 6	syl2anc 415	1 |- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> A ~~ B )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   -->wf 5373   -1-1-onto->wf1o 5376   ~~ cen 7020

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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

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 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-en 7023

This theorem is used by:  f1oeng  7043  enrefg  7050  en2d  7054  en3d  7055  ener  7066  f1imaen2g  7080  cnven  7096  xpcomen  7125  exmidpw2en  7219  xpfi  7239  iccen  10409  nnenom  10871  eqgen  14030