Theorem f1dom2g 6928

Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6930 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
f1dom2g	|- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B )

Proof of Theorem f1dom2g

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1f 5542	. . . . 5 |- ( F : A -1-1-> B -> F : A --> B )
2		fex2 5503	. . . . 5 |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V )
3	1, 2	syl3an1 1306	. . . 4 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> F e. _V )
4	3	3coml 1236	. . 3 |- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> F e. _V )
5		simp3 1025	. . 3 |- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> F : A -1-1-> B )
6		f1eq1 5537	. . . 4 |- ( f = F -> ( f : A -1-1-> B <-> F : A -1-1-> B ) )
7	6	spcegv 2894	. . 3 |- ( F e. _V -> ( F : A -1-1-> B -> E. f f : A -1-1-> B ) )
8	4, 5, 7	sylc 62	. 2 |- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> E. f f : A -1-1-> B )
9		brdomg 6918	. . 3 |- ( B e. W -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
10	9	3ad2ant2 1045	. 2 |- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
11	8, 10	mpbird 167	1 |- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1004   E.wex 1540   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   -->wf 5322   -1-1->wf1 5323   ~<_ cdom 6907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-dom 6910

This theorem is referenced by:  ssdomg  6951