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Theorem mptpreima 5166
Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypothesis
Ref Expression
dmmpt2.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
mptpreima  |-  ( `' F " C )  =  { x  e.  A  |  B  e.  C }
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem mptpreima
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dmmpt2.1 . . . . . 6  |-  F  =  ( x  e.  A  |->  B )
2 df-mpt 4079 . . . . . 6  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
31, 2eqtri 2303 . . . . 5  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
43cnveqi 4856 . . . 4  |-  `' F  =  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
5 cnvopab 5083 . . . 4  |-  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }
64, 5eqtri 2303 . . 3  |-  `' F  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }
76imaeq1i 5009 . 2  |-  ( `' F " C )  =  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) } " C )
8 df-ima 4702 . . 3  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) } " C
)  =  ran  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  |`  C )
9 resopab 4996 . . . . 5  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) }  |`  C )  =  { <. y ,  x >.  |  (
y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }
109rneqi 4905 . . . 4  |-  ran  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  |`  C )  =  ran  { <. y ,  x >.  |  ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }
11 ancom 437 . . . . . . . . 9  |-  ( ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) )  <->  ( (
x  e.  A  /\  y  =  B )  /\  y  e.  C
) )
12 anass 630 . . . . . . . . 9  |-  ( ( ( x  e.  A  /\  y  =  B
)  /\  y  e.  C )  <->  ( x  e.  A  /\  (
y  =  B  /\  y  e.  C )
) )
1311, 12bitri 240 . . . . . . . 8  |-  ( ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) )  <->  ( x  e.  A  /\  (
y  =  B  /\  y  e.  C )
) )
1413exbii 1569 . . . . . . 7  |-  ( E. y ( y  e.  C  /\  ( x  e.  A  /\  y  =  B ) )  <->  E. y
( x  e.  A  /\  ( y  =  B  /\  y  e.  C
) ) )
15 19.42v 1846 . . . . . . . 8  |-  ( E. y ( x  e.  A  /\  ( y  =  B  /\  y  e.  C ) )  <->  ( x  e.  A  /\  E. y
( y  =  B  /\  y  e.  C
) ) )
16 df-clel 2279 . . . . . . . . . 10  |-  ( B  e.  C  <->  E. y
( y  =  B  /\  y  e.  C
) )
1716bicomi 193 . . . . . . . . 9  |-  ( E. y ( y  =  B  /\  y  e.  C )  <->  B  e.  C )
1817anbi2i 675 . . . . . . . 8  |-  ( ( x  e.  A  /\  E. y ( y  =  B  /\  y  e.  C ) )  <->  ( x  e.  A  /\  B  e.  C ) )
1915, 18bitri 240 . . . . . . 7  |-  ( E. y ( x  e.  A  /\  ( y  =  B  /\  y  e.  C ) )  <->  ( x  e.  A  /\  B  e.  C ) )
2014, 19bitri 240 . . . . . 6  |-  ( E. y ( y  e.  C  /\  ( x  e.  A  /\  y  =  B ) )  <->  ( x  e.  A  /\  B  e.  C ) )
2120abbii 2395 . . . . 5  |-  { x  |  E. y ( y  e.  C  /\  (
x  e.  A  /\  y  =  B )
) }  =  {
x  |  ( x  e.  A  /\  B  e.  C ) }
22 rnopab 4924 . . . . 5  |-  ran  { <. y ,  x >.  |  ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }  =  { x  |  E. y ( y  e.  C  /\  ( x  e.  A  /\  y  =  B ) ) }
23 df-rab 2552 . . . . 5  |-  { x  e.  A  |  B  e.  C }  =  {
x  |  ( x  e.  A  /\  B  e.  C ) }
2421, 22, 233eqtr4i 2313 . . . 4  |-  ran  { <. y ,  x >.  |  ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }  =  { x  e.  A  |  B  e.  C }
2510, 24eqtri 2303 . . 3  |-  ran  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  |`  C )  =  { x  e.  A  |  B  e.  C }
268, 25eqtri 2303 . 2  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) } " C
)  =  { x  e.  A  |  B  e.  C }
277, 26eqtri 2303 1  |-  ( `' F " C )  =  { x  e.  A  |  B  e.  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547   {copab 4076    e. cmpt 4077   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692
This theorem is referenced by:  mptiniseg  5167  dmmpt  5168  fmpt  5681  suppss2  6073  suppssfv  6074  suppssov1  6075  cantnfreslem  7377  cantnfres  7379  cantnflem1  7391  cantnf  7395  r0weon  7640  compss  8002  infmsup  9732  eqglact  14668  odngen  14888  rrgsupp  16032  psrbagsn  16236  coe1mul2lem2  16345  pjdm  16607  xkoccn  17313  txcnmpt  17318  txdis1cn  17329  pthaus  17332  txkgen  17346  xkoco1cn  17351  xkoco2cn  17352  xkoinjcn  17381  txcon  17383  ptcmplem1  17746  ptcmplem3  17748  ptcmplem4  17749  tmdgsum2  17779  symgtgp  17784  tgpconcompeqg  17794  ghmcnp  17797  tgpt0  17801  divstgpopn  17802  divstgphaus  17805  eltsms  17815  prdsxmslem2  18075  efopn  20005  atansopn  20228  xrlimcnp  20263  lgseisenlem3  20590  lgseisenlem4  20591  suppss2f  23201  uvcff  27240  pwfi2f1o  27260
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-ral 2548  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-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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