Theorem elrnmpo 5955

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 5952	. . 3 |- ran F = { z | E. x e. A E. y e. B z = C }
3	2	eleq2i 2233	. 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 2229	. . . . . 6 |- ( D = C -> ( D e. _V <-> C e. _V ) )
6	4, 5	mpbiri 167	. . . . 5 |- ( D = C -> D e. _V )
7	6	rexlimivw 2579	. . . 4 |- ( E. y e. B D = C -> D e. _V )
8	7	rexlimivw 2579	. . 3 |- ( E. x e. A E. y e. B D = C -> D e. _V )
9		eqeq1 2172	. . . 4 |- ( z = D -> ( z = C <-> D = C ) )
10	9	2rexbidv 2491	. . 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 2878	. 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 183	1 |- ( D e. ran F <-> E. x e. A E. y e. B D = C )

Syntax hints:   <-> wb 104   = wceq 1343   e. wcel 2136   {cab 2151   E.wrex 2445   _Vcvv 2726   ran crn 4605   e. cmpo 5844

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615  df-oprab 5846  df-mpo 5847

This theorem is referenced by: (None)