Theorem dom2 6669

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 NM, 26-Oct-2003.)

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
dom2	|- ( B e. V -> A ~<_ B )

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

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

Proof of Theorem dom2

Step	Hyp	Ref	Expression
1		eqid 2139	. 2 |- A = A
2		dom2.1	. . . 4 |- ( x e. A -> C e. B )
3	2	a1i 9	. . 3 |- ( A = A -> ( x e. A -> C e. B ) )
4		dom2.2	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) )
5	4	a1i 9	. . 3 |- ( A = A -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )
6	3, 5	dom2d 6667	. 2 |- ( A = A -> ( B e. V -> A ~<_ B ) )
7	1, 6	ax-mp 5	1 |- ( B e. V -> A ~<_ B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   class class class wbr 3929   ~<_ cdom 6633

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-dom 6636

This theorem is referenced by:  infpwfidom  7054  tgdom  12241