Theorem f1oen4g 6901

Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 6906 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)

Assertion

Ref	Expression
f1oen4g	|- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-onto-> B ) -> A ~~ B )

Proof of Theorem f1oen4g

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oeq1 5559	. . . . 5 |- ( f = F -> ( f : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
2	1	spcegv 2891	. . . 4 |- ( F e. V -> ( F : A -1-1-onto-> B -> E. f f : A -1-1-onto-> B ) )
3	2	imp 124	. . 3 |- ( ( F e. V /\ F : A -1-1-onto-> B ) -> E. f f : A -1-1-onto-> B )
4	3	3ad2antl1 1183	. 2 |- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-onto-> B ) -> E. f f : A -1-1-onto-> B )
5		breng 6892	. . . 4 |- ( ( A e. W /\ B e. X ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )
6	5	3adant1 1039	. . 3 |- ( ( F e. V /\ A e. W /\ B e. X ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )
7	6	adantr 276	. 2 |- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-onto-> B ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )
8	4, 7	mpbird 167	1 |- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-onto-> B ) -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   E.wex 1538   e. wcel 2200   class class class wbr 4082   -1-1-onto->wf1o 5316   ~~ cen 6883

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-en 6886

This theorem is referenced by: (None)