Theorem f1oen4g 6991

Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 6996 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 5602	. . . . 5 |- ( f = F -> ( f : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
2	1	spcegv 2905	. . . 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 1186	. 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 6982	. . . 4 |- ( ( A e. W /\ B e. X ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )
6	5	3adant1 1042	. . 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 1005   E.wex 1541   e. wcel 2203   class class class wbr 4109   -1-1-onto->wf1o 5351   ~~ cen 6973

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-en 6976

This theorem is referenced by: (None)