Theorem elrnmpo 6006

Description: Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
rngop.1	|- F = ( x e. A , y e. B |-> C )
elrnmpo.1	|- C e. _V

Assertion

Ref	Expression
elrnmpo	|- ( D e. ran F <-> E. x e. A E. y e. B D = C )

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

Allowed substitution hints:   A( x)   B( x, y)   C( x, y)   F( x, y)

Proof of Theorem elrnmpo

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngop.1	. . . 4 |- F = ( x e. A , y e. B |-> C )
2	1	rnmpo 6003	. . 3 |- ran F = { z | E. x e. A E. y e. B z = C }
3	2	eleq2i 2256	. 2 |- ( D e. ran F <-> D e. { z | E. x e. A E. y e. B z = C } )
4		elrnmpo.1	. . . . . 6 |- C e. _V
5		eleq1 2252	. . . . . 6 |- ( D = C -> ( D e. _V <-> C e. _V ) )
6	4, 5	mpbiri 168	. . . . 5 |- ( D = C -> D e. _V )
7	6	rexlimivw 2603	. . . 4 |- ( E. y e. B D = C -> D e. _V )
8	7	rexlimivw 2603	. . 3 |- ( E. x e. A E. y e. B D = C -> D e. _V )
9		eqeq1 2196	. . . 4 |- ( z = D -> ( z = C <-> D = C ) )
10	9	2rexbidv 2515	. . 3 |- ( z = D -> ( E. x e. A E. y e. B z = C <-> E. x e. A E. y e. B D = C ) )
11	8, 10	elab3 2904	. 2 |- ( D e. { z | E. x e. A E. y e. B z = C } <-> E. x e. A E. y e. B D = C )
12	3, 11	bitri 184	1 |- ( D e. ran F <-> E. x e. A E. y e. B D = C )

Syntax hints:   <-> wb 105   = wceq 1364   e. wcel 2160   {cab 2175   E.wrex 2469   _Vcvv 2752   ran crn 4642   e. cmpo 5894

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-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-cnv 4649  df-dm 4651  df-rn 4652  df-oprab 5896  df-mpo 5897

This theorem is referenced by: (None)