Theorem f1dom2g 6782

Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6784 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 5440	. . . . 5 |- ( F : A -1-1-> B -> F : A --> B )
2		fex2 5403	. . . . 5 |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V )
3	1, 2	syl3an1 1282	. . . 4 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> F e. _V )
4	3	3coml 1212	. . 3 |- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> F e. _V )
5		simp3 1001	. . 3 |- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> F : A -1-1-> B )
6		f1eq1 5435	. . . 4 |- ( f = F -> ( f : A -1-1-> B <-> F : A -1-1-> B ) )
7	6	spcegv 2840	. . 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 6774	. . 3 |- ( B e. W -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
10	9	3ad2ant2 1021	. 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 980   E.wex 1503   e. wcel 2160   _Vcvv 2752   class class class wbr 4018   -->wf 5231   -1-1->wf1 5232   ~<_ cdom 6765

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-dom 6768

This theorem is referenced by:  ssdomg  6804