Theorem elrnmpog 6116

Description: Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

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

Assertion

Ref	Expression
elrnmpog	|- ( D e. V -> ( 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)   V( x, y)

Proof of Theorem elrnmpog

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2236	. . 3 |- ( z = D -> ( z = C <-> D = C ) )
2	1	2rexbidv 2555	. 2 |- ( z = D -> ( E. x e. A E. y e. B z = C <-> E. x e. A E. y e. B D = C ) )
3		rngop.1	. . 3 |- F = ( x e. A , y e. B |-> C )
4	3	rnmpo 6114	. 2 |- ran F = { z | E. x e. A E. y e. B z = C }
5	2, 4	elab2g 2950	1 |- ( D e. V -> ( D e. ran F <-> E. x e. A E. y e. B D = C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   ran crn 4719   e. cmpo 6002

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-cnv 4726  df-dm 4728  df-rn 4729  df-oprab 6004  df-mpo 6005

This theorem is referenced by:  txopn  14933  xmettxlem  15177  xmettx  15178