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Theorem rnmpt2 5954
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 5863 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
31, 2eqtri 2303 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
43rneqi 4905 . 2  |-  ran  F  =  ran  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
5 rnoprab2 5931 . 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 2303 1  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   ran crn 4690   {coprab 5859    e. cmpt2 5860
This theorem is referenced by:  elrnmpt2g  5956  elrnmpt2  5957  ralrnmpt2  5958  dffi3  7184  ixpiunwdom  7305  qnnen  12492  txuni2  17260  txbas  17262  xkobval  17281  xkoopn  17284  txrest  17325  ptrescn  17333  tx1stc  17344  xkoptsub  17348  xkopt  17349  xkococn  17354  ptcmplem4  17749  met2ndci  18068  i1fadd  19050  i1fmul  19051  rnmpt2ss  23239  cnre2csqima  23295  cbcpcp  25162  eldiophb  26836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700  df-oprab 5862  df-mpt2 5863
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