Theorem dom3d 6990

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 6988	. . . . 5 |- ( ph -> ( x e. A |-> C ) : A -1-1-> B )
4		f1f 5551	. . . . 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 5511	. . . 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 1274	. . 3 |- ( ph -> ( x e. A |-> C ) e. _V )
10		f1eq1 5546	. . . 4 |- ( z = ( x e. A |-> C ) -> ( z : A -1-1-> B <-> ( x e. A |-> C ) : A -1-1-> B ) )
11	10	spcegv 2895	. . 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 6962	. . 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 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   class class class wbr 4093   |-> cmpt 4155   -->wf 5329   -1-1->wf1 5330   ~<_ cdom 6951

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fv 5341  df-dom 6954

This theorem is referenced by:  dom3  6992  xpdom2  7058  fopwdom  7065  nninfinf  10751