Theorem rnmpo 6127

Description: The range of an operation given by the maps-to notation. (Contributed by FL, 20-Jun-2011.)

Hypothesis

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

Assertion

Ref	Expression
rnmpo	|- ran F = { z | E. x e. A E. y e. B z = C }

Distinct variable groups:   y, z, A   z, B   z, C   z, F   x, y, z

Allowed substitution hints:   A( x)   B( x, y)   C( x, y)   F( x, y)

Proof of Theorem rnmpo

Step	Hyp	Ref	Expression
1		rngop.1	. . . 4 |- F = ( x e. A , y e. B |-> C )
2		df-mpo 6018	. . . 4 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
3	1, 2	eqtri 2250	. . 3 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
4	3	rneqi 4958	. 2 |- ran F = ran { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
5		rnoprab2 6100	. 2 |- ran { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) } = { z | E. x e. A E. y e. B z = C }
6	4, 5	eqtri 2250	1 |- ran F = { z | E. x e. A E. y e. B z = C }

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   ran crn 4724   {coprab 6014   e. cmpo 6015

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 4205  ax-pow 4262  ax-pr 4297

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 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-cnv 4731  df-dm 4733  df-rn 4734  df-oprab 6017  df-mpo 6018

This theorem is referenced by:  elrnmpog  6129  elrnmpo  6130  ralrnmpo  6131  rexrnmpo  6132  mpoexw  6373  txuni2  14970  txbas  14972  txrest  14990