Theorem f1dom4g 7029

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 7034 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 5588	. . . . 5 |- ( f = F -> ( f : A -1-1-> B <-> F : A -1-1-> B ) )
2	1	spcegv 2913	. . . 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 1190	. 2 |- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-> B ) -> E. f f : A -1-1-> B )
5		brdom2g 7021	. . . 4 |- ( ( A e. W /\ B e. X ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
6	5	3adant1 1046	. . 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 1009   E.wex 1545   e. wcel 2209   class class class wbr 4125   -1-1->wf1 5369   ~<_ cdom 7011

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 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 4244  ax-pow 4306  ax-pr 4341

This theorem 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-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-dom 7014

This theorem is referenced by:  domssr  7054