Theorem elrnmpt2 5772

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 )
elrnmpt2.1	|- C e. _V

Assertion

Ref	Expression
elrnmpt2	|- ( 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 elrnmpt2

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	rnmpt2 5769	. . 3 |- ran F = { z | E. x e. A E. y e. B z = C }
3	2	eleq2i 2155	. 2 |- ( D e. ran F <-> D e. { z | E. x e. A E. y e. B z = C } )
4		elrnmpt2.1	. . . . . 6 |- C e. _V
5		eleq1 2151	. . . . . 6 |- ( D = C -> ( D e. _V <-> C e. _V ) )
6	4, 5	mpbiri 167	. . . . 5 |- ( D = C -> D e. _V )
7	6	rexlimivw 2486	. . . 4 |- ( E. y e. B D = C -> D e. _V )
8	7	rexlimivw 2486	. . 3 |- ( E. x e. A E. y e. B D = C -> D e. _V )
9		eqeq1 2095	. . . 4 |- ( z = D -> ( z = C <-> D = C ) )
10	9	2rexbidv 2404	. . 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 2768	. 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 1290   e. wcel 1439   {cab 2075   E.wrex 2361   _Vcvv 2620   ran crn 4453   |-> cmpt2 5668

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-cnv 4460  df-dm 4462  df-rn 4463  df-oprab 5670  df-mpt2 5671

This theorem is referenced by: (None)