Theorem dom3d 6481

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 6479	. . . . 5 |- ( ph -> ( x e. A |-> C ) : A -1-1-> B )
4		f1f 5210	. . . . 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 5173	. . . 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 1174	. . 3 |- ( ph -> ( x e. A |-> C ) e. _V )
10		f1eq1 5205	. . . 4 |- ( z = ( x e. A |-> C ) -> ( z : A -1-1-> B <-> ( x e. A |-> C ) : A -1-1-> B ) )
11	10	spcegv 2707	. . 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 6455	. . 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 1289   E.wex 1426   e. wcel 1438   _Vcvv 2619   class class class wbr 3843   |-> cmpt 3897   -->wf 5006   -1-1->wf1 5007   ~<_ cdom 6446

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fv 5018  df-dom 6449

This theorem is referenced by:  dom3  6483  xpdom2  6537  fopwdom  6542