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Theorem mptpreima 5304
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 4210 . . . . . 6  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
31, 2eqtri 2408 . . . . 5  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
43cnveqi 4988 . . . 4  |-  `' F  =  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
5 cnvopab 5215 . . . 4  |-  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }
64, 5eqtri 2408 . . 3  |-  `' F  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }
76imaeq1i 5141 . 2  |-  ( `' F " C )  =  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) } " C )
8 df-ima 4832 . . 3  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) } " C
)  =  ran  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  |`  C )
9 resopab 5128 . . . . 5  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) }  |`  C )  =  { <. y ,  x >.  |  (
y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }
109rneqi 5037 . . . 4  |-  ran  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  |`  C )  =  ran  { <. y ,  x >.  |  ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }
11 ancom 438 . . . . . . . . 9  |-  ( ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) )  <->  ( (
x  e.  A  /\  y  =  B )  /\  y  e.  C
) )
12 anass 631 . . . . . . . . 9  |-  ( ( ( x  e.  A  /\  y  =  B
)  /\  y  e.  C )  <->  ( x  e.  A  /\  (
y  =  B  /\  y  e.  C )
) )
1311, 12bitri 241 . . . . . . . 8  |-  ( ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) )  <->  ( x  e.  A  /\  (
y  =  B  /\  y  e.  C )
) )
1413exbii 1589 . . . . . . 7  |-  ( E. y ( y  e.  C  /\  ( x  e.  A  /\  y  =  B ) )  <->  E. y
( x  e.  A  /\  ( y  =  B  /\  y  e.  C
) ) )
15 19.42v 1917 . . . . . . . 8  |-  ( E. y ( x  e.  A  /\  ( y  =  B  /\  y  e.  C ) )  <->  ( x  e.  A  /\  E. y
( y  =  B  /\  y  e.  C
) ) )
16 df-clel 2384 . . . . . . . . . 10  |-  ( B  e.  C  <->  E. y
( y  =  B  /\  y  e.  C
) )
1716bicomi 194 . . . . . . . . 9  |-  ( E. y ( y  =  B  /\  y  e.  C )  <->  B  e.  C )
1817anbi2i 676 . . . . . . . 8  |-  ( ( x  e.  A  /\  E. y ( y  =  B  /\  y  e.  C ) )  <->  ( x  e.  A  /\  B  e.  C ) )
1915, 18bitri 241 . . . . . . 7  |-  ( E. y ( x  e.  A  /\  ( y  =  B  /\  y  e.  C ) )  <->  ( x  e.  A  /\  B  e.  C ) )
2014, 19bitri 241 . . . . . 6  |-  ( E. y ( y  e.  C  /\  ( x  e.  A  /\  y  =  B ) )  <->  ( x  e.  A  /\  B  e.  C ) )
2120abbii 2500 . . . . 5  |-  { x  |  E. y ( y  e.  C  /\  (
x  e.  A  /\  y  =  B )
) }  =  {
x  |  ( x  e.  A  /\  B  e.  C ) }
22 rnopab 5056 . . . . 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 2659 . . . . 5  |-  { x  e.  A  |  B  e.  C }  =  {
x  |  ( x  e.  A  /\  B  e.  C ) }
2421, 22, 233eqtr4i 2418 . . . 4  |-  ran  { <. y ,  x >.  |  ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }  =  { x  e.  A  |  B  e.  C }
2510, 24eqtri 2408 . . 3  |-  ran  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  |`  C )  =  { x  e.  A  |  B  e.  C }
268, 25eqtri 2408 . 2  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) } " C
)  =  { x  e.  A  |  B  e.  C }
277, 26eqtri 2408 1  |-  ( `' F " C )  =  { x  e.  A  |  B  e.  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374   {crab 2654   {copab 4207    e. cmpt 4208   `'ccnv 4818   ran crn 4820    |` cres 4821   "cima 4822
This theorem is referenced by:  mptiniseg  5305  dmmpt  5306  fmpt  5830  suppss2  6240  suppssfv  6241  suppssov1  6242  cantnfreslem  7565  cantnfres  7567  cantnflem1  7579  cantnf  7583  r0weon  7828  compss  8190  infmsup  9919  eqglact  14919  odngen  15139  rrgsupp  16279  psrbagsn  16483  coe1mul2lem2  16589  pjdm  16858  xkoccn  17573  txcnmpt  17578  txdis1cn  17589  pthaus  17592  txkgen  17606  xkoco1cn  17611  xkoco2cn  17612  xkoinjcn  17641  txcon  17643  imasnopn  17644  imasncld  17645  imasncls  17646  ptcmplem1  18005  ptcmplem3  18007  ptcmplem4  18008  tmdgsum2  18048  symgtgp  18053  tgpconcompeqg  18063  ghmcnp  18066  tgpt0  18070  divstgpopn  18071  divstgphaus  18074  eltsms  18084  prdsxmslem2  18450  efopn  20417  atansopn  20640  xrlimcnp  20675  lgseisenlem3  21003  lgseisenlem4  21004  suppss2f  23893  itg2addnclem2  25959  itgaddnclem2  25965  iblabsnclem  25969  uvcff  26910  pwfi2f1o  26930
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-mpt 4210  df-xp 4825  df-rel 4826  df-cnv 4827  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832
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