Theorem dom3 6279

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. C and D can be read C ( x ) and D ( y ), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)

Hypotheses

Ref	Expression
dom2.1	|- ( x e. A -> C e. B )
dom2.2	|- ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) )

Assertion

Ref	Expression
dom3	|- ( ( A e. V /\ B e. W ) -> A ~<_ B )

Distinct variable groups:   x, y, A   x, B, y   y, C   x, D   x, V, y   x, W, y

Allowed substitution hints:   C( x)   D( y)

Proof of Theorem dom3

Step	Hyp	Ref	Expression
1		dom2.1	. . 3 |- ( x e. A -> C e. B )
2	1	a1i 9	. 2 |- ( ( A e. V /\ B e. W ) -> ( x e. A -> C e. B ) )
3		dom2.2	. . 3 |- ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) )
4	3	a1i 9	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )
5		simpl 107	. 2 |- ( ( A e. V /\ B e. W ) -> A e. V )
6		simpr 108	. 2 |- ( ( A e. V /\ B e. W ) -> B e. W )
7	2, 4, 5, 6	dom3d 6277	1 |- ( ( A e. V /\ B e. W ) -> A ~<_ B )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   class class class wbr 3785   ~<_ cdom 6243

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fv 4930  df-dom 6246

This theorem is referenced by: (None)