Theorem f1dom4g 6857

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 6862 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 5488	. . . . 5 |- ( f = F -> ( f : A -1-1-> B <-> F : A -1-1-> B ) )
2	1	spcegv 2865	. . . 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 1162	. 2 |- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-> B ) -> E. f f : A -1-1-> B )
5		brdom2g 6849	. . . 4 |- ( ( A e. W /\ B e. X ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
6	5	3adant1 1018	. . 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 981   E.wex 1516   e. wcel 2177   class class class wbr 4051   -1-1->wf1 5277   ~<_ cdom 6839

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-dom 6842

This theorem is referenced by:  domssr  6882