Theorem elrnmpt2 5614

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 5611	. . 3 |- ran F = { z | E. x e. A E. y e. B z = C }
3	2	eleq2i 2104	. 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 2100	. . . . . 6 |- ( D = C -> ( D e. _V <-> C e. _V ) )
6	4, 5	mpbiri 157	. . . . 5 |- ( D = C -> D e. _V )
7	6	rexlimivw 2429	. . . 4 |- ( E. y e. B D = C -> D e. _V )
8	7	rexlimivw 2429	. . 3 |- ( E. x e. A E. y e. B D = C -> D e. _V )
9		eqeq1 2046	. . . 4 |- ( z = D -> ( z = C <-> D = C ) )
10	9	2rexbidv 2349	. . 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 2694	. 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 173	1 |- ( D e. ran F <-> E. x e. A E. y e. B D = C )

Syntax hints:   <-> wb 98   = wceq 1243   e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557   ran crn 4346   |-> cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356  df-oprab 5516  df-mpt2 5517

This theorem is referenced by: (None)