Theorem dom3d 6946

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)

Hypotheses

Ref	Expression
dom2d.1	|- ( ph -> ( x e. A -> C e. B ) )
dom2d.2	|- ( ph -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )
dom3d.3	|- ( ph -> A e. V )
dom3d.4	|- ( ph -> B e. W )

Assertion

Ref	Expression
dom3d	|- ( ph -> A ~<_ B )

Distinct variable groups:   x, y, A   x, B, y   y, C   x, D   ph, x, y

Allowed substitution hints:   C( x)   D( y)   V( x, y)   W( x, y)

Proof of Theorem dom3d

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dom2d.1	. . . . . 6 |- ( ph -> ( x e. A -> C e. B ) )
2		dom2d.2	. . . . . 6 |- ( ph -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )
3	1, 2	dom2lem 6944	. . . . 5 |- ( ph -> ( x e. A |-> C ) : A -1-1-> B )
4		f1f 5542	. . . . 5 |- ( ( x e. A |-> C ) : A -1-1-> B -> ( x e. A |-> C ) : A --> B )
5	3, 4	syl 14	. . . 4 |- ( ph -> ( x e. A |-> C ) : A --> B )
6		dom3d.3	. . . 4 |- ( ph -> A e. V )
7		dom3d.4	. . . 4 |- ( ph -> B e. W )
8		fex2 5503	. . . 4 |- ( ( ( x e. A |-> C ) : A --> B /\ A e. V /\ B e. W ) -> ( x e. A |-> C ) e. _V )
9	5, 6, 7, 8	syl3anc 1273	. . 3 |- ( ph -> ( x e. A |-> C ) e. _V )
10		f1eq1 5537	. . . 4 |- ( z = ( x e. A |-> C ) -> ( z : A -1-1-> B <-> ( x e. A |-> C ) : A -1-1-> B ) )
11	10	spcegv 2894	. . 3 |- ( ( x e. A |-> C ) e. _V -> ( ( x e. A |-> C ) : A -1-1-> B -> E. z z : A -1-1-> B ) )
12	9, 3, 11	sylc 62	. 2 |- ( ph -> E. z z : A -1-1-> B )
13		brdomg 6918	. . 3 |- ( B e. W -> ( A ~<_ B <-> E. z z : A -1-1-> B ) )
14	7, 13	syl 14	. 2 |- ( ph -> ( A ~<_ B <-> E. z z : A -1-1-> B ) )
15	12, 14	mpbird 167	1 |- ( ph -> A ~<_ B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   |-> cmpt 4150   -->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-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  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-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fv 5334  df-dom 6910

This theorem is referenced by:  dom3  6948  xpdom2  7014  fopwdom  7021  nninfinf  10704