Theorem dom3d 6752

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 6750	. . . . 5 |- ( ph -> ( x e. A |-> C ) : A -1-1-> B )
4		f1f 5403	. . . . 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 5366	. . . 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 1233	. . 3 |- ( ph -> ( x e. A |-> C ) e. _V )
10		f1eq1 5398	. . . 4 |- ( z = ( x e. A |-> C ) -> ( z : A -1-1-> B <-> ( x e. A |-> C ) : A -1-1-> B ) )
11	10	spcegv 2818	. . 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 6726	. . 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 166	1 |- ( ph -> A ~<_ B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   |-> cmpt 4050   -->wf 5194   -1-1->wf1 5195   ~<_ cdom 6717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fv 5206  df-dom 6720

This theorem is referenced by:  dom3  6754  xpdom2  6809  fopwdom  6814