Theorem dom3 6742

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 108	. 2 |- ( ( A e. V /\ B e. W ) -> A e. V )
6		simpr 109	. 2 |- ( ( A e. V /\ B e. W ) -> B e. W )
7	2, 4, 5, 6	dom3d 6740	1 |- ( ( A e. V /\ B e. W ) -> A ~<_ B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   class class class wbr 3982   ~<_ cdom 6705

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fv 5196  df-dom 6708

This theorem is referenced by: (None)