Theorem rnmpt2 6770

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
rnmpt2	|- 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 rnmpt2

Step	Hyp	Ref	Expression
1		rngop.1	. . . 4 |- F = ( x e. A , y e. B |-> C )
2		df-mpt2 6655	. . . 4 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
3	1, 2	eqtri 2644	. . 3 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
4	3	rneqi 5352	. 2 |- ran F = ran { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
5		rnoprab2 6744	. 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 2644	1 |- ran F = { z | E. x e. A E. y e. B z = C }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   ran crn 5115   {coprab 6651   |-> 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:  elrnmpt2g  6772  elrnmpt2  6773  ralrnmpt2  6775  dffi3  8337  ixpiunwdom  8496  qnnen  14942  txuni2  21368  txbas  21370  xkobval  21389  xkoopn  21392  txrest  21434  ptrescn  21442  tx1stc  21453  xkoptsub  21457  xkopt  21458  xkococn  21463  ptcmplem4  21859  met2ndci  22327  i1fadd  23462  i1fmul  23463  rnmpt2ss  29473  cnre2csqima  29957  qqhval2  30026  scutf  31919  icoreresf  33200  ptrest  33408  eldiophb  37320