Theorem elrnmpo 6134

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 6131	. . 3 |- ran F = { z | E. x e. A E. y e. B z = C }
3	2	eleq2i 2298	. 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 2294	. . . . . 6 |- ( D = C -> ( D e. _V <-> C e. _V ) )
6	4, 5	mpbiri 168	. . . . 5 |- ( D = C -> D e. _V )
7	6	rexlimivw 2646	. . . 4 |- ( E. y e. B D = C -> D e. _V )
8	7	rexlimivw 2646	. . 3 |- ( E. x e. A E. y e. B D = C -> D e. _V )
9		eqeq1 2238	. . . 4 |- ( z = D -> ( z = C <-> D = C ) )
10	9	2rexbidv 2557	. . 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 2958	. 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 1397   e. wcel 2202   {cab 2217   E.wrex 2511   _Vcvv 2802   ran crn 4726   e. cmpo 6019

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-cnv 4733  df-dm 4735  df-rn 4736  df-oprab 6021  df-mpo 6022

This theorem is referenced by: (None)