Theorem dom3 6948

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 109	. 2 |- ( ( A e. V /\ B e. W ) -> A e. V )
6		simpr 110	. 2 |- ( ( A e. V /\ B e. W ) -> B e. W )
7	2, 4, 5, 6	dom3d 6946	1 |- ( ( A e. V /\ B e. W ) -> A ~<_ B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088   ~<_ cdom 6907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fv 5334  df-dom 6910

This theorem is referenced by: (None)