Theorem dom3 7999

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 11	. 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 11	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )
5		simpl 473	. 2 |- ( ( A e. V /\ B e. W ) -> A e. V )
6		simpr 477	. 2 |- ( ( A e. V /\ B e. W ) -> B e. W )
7	2, 4, 5, 6	dom3d 7997	1 |- ( ( A e. V /\ B e. W ) -> A ~<_ B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ~<_ cdom 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-dom 7957

This theorem is referenced by:  canth2  8113  limenpsi  8135