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Theorem rnmpt2 6143
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 6049 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
31, 2eqtri 2428 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
43rneqi 5059 . 2  |-  ran  F  =  ran  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
5 rnoprab2 6120 . 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 2428 1  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2394   E.wrex 2671   ran crn 4842   {coprab 6045    e. cmpt2 6046
This theorem is referenced by:  elrnmpt2g  6145  elrnmpt2  6146  ralrnmpt2  6147  dffi3  7398  ixpiunwdom  7519  qnnen  12772  txuni2  17554  txbas  17556  xkobval  17575  xkoopn  17578  txrest  17620  ptrescn  17628  tx1stc  17639  xkoptsub  17643  xkopt  17644  xkococn  17649  ptcmplem4  18043  met2ndci  18509  i1fadd  19544  i1fmul  19545  rnmpt2ss  24043  cnre2csqima  24266  qqhval2  24323  eldiophb  26709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-cnv 4849  df-dm 4851  df-rn 4852  df-oprab 6048  df-mpt2 6049
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