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Theorem rnmpt2 6183
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
StepHypRef Expression
1 rngop.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
2 df-mpt2 6089 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
31, 2eqtri 2458 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
43rneqi 5099 . 2  |-  ran  F  =  ran  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
5 rnoprab2 6160 . 2  |-  ran  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
64, 5eqtri 2458 1  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   E.wrex 2708   ran crn 4882   {coprab 6085    e. cmpt2 6086
This theorem is referenced by:  elrnmpt2g  6185  elrnmpt2  6186  ralrnmpt2  6187  dffi3  7439  ixpiunwdom  7562  qnnen  12818  txuni2  17602  txbas  17604  xkobval  17623  xkoopn  17626  txrest  17668  ptrescn  17676  tx1stc  17687  xkoptsub  17691  xkopt  17692  xkococn  17697  ptcmplem4  18091  met2ndci  18557  i1fadd  19590  i1fmul  19591  rnmpt2ss  24091  cnre2csqima  24314  qqhval2  24371  eldiophb  26829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-cnv 4889  df-dm 4891  df-rn 4892  df-oprab 6088  df-mpt2 6089
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