Theorem dom3d 6343

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 6341	. . . . 5 |- ( ph -> ( x e. A |-> C ) : A -1-1-> B )
4		f1f 5143	. . . . 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 5110	. . . 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 1170	. . 3 |- ( ph -> ( x e. A |-> C ) e. _V )
10		f1eq1 5138	. . . 4 |- ( z = ( x e. A |-> C ) -> ( z : A -1-1-> B <-> ( x e. A |-> C ) : A -1-1-> B ) )
11	10	spcegv 2695	. . 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 61	. 2 |- ( ph -> E. z z : A -1-1-> B )
13		brdomg 6317	. . 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 165	1 |- ( ph -> A ~<_ B )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   E.wex 1422   e. wcel 1434   _Vcvv 2610   class class class wbr 3805   |-> cmpt 3859   -->wf 4948   -1-1->wf1 4949   ~<_ cdom 6308

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fv 4960  df-dom 6311

This theorem is referenced by:  dom3  6345  xpdom2  6397  fopwdom  6402