Theorem f1dom4g 7005

Description: The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 7010 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)

Assertion

Ref	Expression
f1dom4g	|- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-> B ) -> A ~<_ B )

Proof of Theorem f1dom4g

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1eq1 5573	. . . . 5 |- ( f = F -> ( f : A -1-1-> B <-> F : A -1-1-> B ) )
2	1	spcegv 2907	. . . 4 |- ( F e. V -> ( F : A -1-1-> B -> E. f f : A -1-1-> B ) )
3	2	imp 124	. . 3 |- ( ( F e. V /\ F : A -1-1-> B ) -> E. f f : A -1-1-> B )
4	3	3ad2antl1 1186	. 2 |- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-> B ) -> E. f f : A -1-1-> B )
5		brdom2g 6997	. . . 4 |- ( ( A e. W /\ B e. X ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
6	5	3adant1 1042	. . 3 |- ( ( F e. V /\ A e. W /\ B e. X ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
7	6	adantr 276	. 2 |- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-> B ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
8	4, 7	mpbird 167	1 |- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-> B ) -> A ~<_ B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   E.wex 1541   e. wcel 2205   class class class wbr 4114   -1-1->wf1 5354   ~<_ cdom 6987

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-dom 6990

This theorem is referenced by:  domssr  7030