Theorem dom2 6831

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 2193	. 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 6829	. 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 104   <-> wb 105   = wceq 1364   e. wcel 2164   class class class wbr 4030   ~<_ cdom 6795

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dom 6798

This theorem is referenced by:  infpwfidom  7260  tgdom  14251