Theorem elrnmpt2g 6772

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
elrnmpt2g	|- ( 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 elrnmpt2g

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . 3 |- ( z = D -> ( z = C <-> D = C ) )
2	1	2rexbidv 3057	. 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	rnmpt2 6770	. 2 |- ran F = { z | E. x e. A E. y e. B z = C }
5	2, 4	elab2g 3353	1 |- ( D e. V -> ( D e. ran F <-> E. x e. A E. y e. B D = C ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   ran crn 5115   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  ordtbas2  20995  txopn  21405  tgisline  25522  elsx  30257  smflimlem6  40984