Theorem elrnmpog 6057

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 2211	. . 3 |- ( z = D -> ( z = C <-> D = C ) )
2	1	2rexbidv 2530	. 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 6055	. 2 |- ran F = { z | E. x e. A E. y e. B z = C }
5	2, 4	elab2g 2919	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 1372   e. wcel 2175   E.wrex 2484   ran crn 4675   e. cmpo 5945

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-cnv 4682  df-dm 4684  df-rn 4685  df-oprab 5947  df-mpo 5948

This theorem is referenced by:  txopn  14708  xmettxlem  14952  xmettx  14953