Theorem dom2d 6864

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised 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 ) ) )

Assertion

Ref	Expression
dom2d	|- ( ph -> ( B e. R -> 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)   R( x, y)

Proof of Theorem dom2d

Step	Hyp	Ref	Expression
1		dom2d.1	. . 3 |- ( ph -> ( x e. A -> C e. B ) )
2		dom2d.2	. . 3 |- ( ph -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )
3	1, 2	dom2lem 6863	. 2 |- ( ph -> ( x e. A |-> C ) : A -1-1-> B )
4		f1domg 6849	. 2 |- ( B e. R -> ( ( x e. A |-> C ) : A -1-1-> B -> A ~<_ B ) )
5	3, 4	syl5com 29	1 |- ( ph -> ( B e. R -> A ~<_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   class class class wbr 4044   |-> cmpt 4105   -1-1->wf1 5268   ~<_ cdom 6826

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-dom 6829

This theorem is referenced by:  dom2  6866