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Theorem mptpreima 5366
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 4271 . . . . . 6  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
31, 2eqtri 2458 . . . . 5  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
43cnveqi 5050 . . . 4  |-  `' F  =  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
5 cnvopab 5277 . . . 4  |-  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }
64, 5eqtri 2458 . . 3  |-  `' F  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }
76imaeq1i 5203 . 2  |-  ( `' F " C )  =  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) } " C )
8 df-ima 4894 . . 3  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) } " C
)  =  ran  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  |`  C )
9 resopab 5190 . . . . 5  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) }  |`  C )  =  { <. y ,  x >.  |  (
y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }
109rneqi 5099 . . . 4  |-  ran  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  |`  C )  =  ran  { <. y ,  x >.  |  ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }
11 ancom 439 . . . . . . . . 9  |-  ( ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) )  <->  ( (
x  e.  A  /\  y  =  B )  /\  y  e.  C
) )
12 anass 632 . . . . . . . . 9  |-  ( ( ( x  e.  A  /\  y  =  B
)  /\  y  e.  C )  <->  ( x  e.  A  /\  (
y  =  B  /\  y  e.  C )
) )
1311, 12bitri 242 . . . . . . . 8  |-  ( ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) )  <->  ( x  e.  A  /\  (
y  =  B  /\  y  e.  C )
) )
1413exbii 1593 . . . . . . 7  |-  ( E. y ( y  e.  C  /\  ( x  e.  A  /\  y  =  B ) )  <->  E. y
( x  e.  A  /\  ( y  =  B  /\  y  e.  C
) ) )
15 19.42v 1929 . . . . . . . 8  |-  ( E. y ( x  e.  A  /\  ( y  =  B  /\  y  e.  C ) )  <->  ( x  e.  A  /\  E. y
( y  =  B  /\  y  e.  C
) ) )
16 df-clel 2434 . . . . . . . . . 10  |-  ( B  e.  C  <->  E. y
( y  =  B  /\  y  e.  C
) )
1716bicomi 195 . . . . . . . . 9  |-  ( E. y ( y  =  B  /\  y  e.  C )  <->  B  e.  C )
1817anbi2i 677 . . . . . . . 8  |-  ( ( x  e.  A  /\  E. y ( y  =  B  /\  y  e.  C ) )  <->  ( x  e.  A  /\  B  e.  C ) )
1915, 18bitri 242 . . . . . . 7  |-  ( E. y ( x  e.  A  /\  ( y  =  B  /\  y  e.  C ) )  <->  ( x  e.  A  /\  B  e.  C ) )
2014, 19bitri 242 . . . . . 6  |-  ( E. y ( y  e.  C  /\  ( x  e.  A  /\  y  =  B ) )  <->  ( x  e.  A  /\  B  e.  C ) )
2120abbii 2550 . . . . 5  |-  { x  |  E. y ( y  e.  C  /\  (
x  e.  A  /\  y  =  B )
) }  =  {
x  |  ( x  e.  A  /\  B  e.  C ) }
22 rnopab 5118 . . . . 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 2716 . . . . 5  |-  { x  e.  A  |  B  e.  C }  =  {
x  |  ( x  e.  A  /\  B  e.  C ) }
2421, 22, 233eqtr4i 2468 . . . 4  |-  ran  { <. y ,  x >.  |  ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }  =  { x  e.  A  |  B  e.  C }
2510, 24eqtri 2458 . . 3  |-  ran  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  |`  C )  =  { x  e.  A  |  B  e.  C }
268, 25eqtri 2458 . 2  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) } " C
)  =  { x  e.  A  |  B  e.  C }
277, 26eqtri 2458 1  |-  ( `' F " C )  =  { x  e.  A  |  B  e.  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   {crab 2711   {copab 4268    e. cmpt 4269   `'ccnv 4880   ran crn 4882    |` cres 4883   "cima 4884
This theorem is referenced by:  mptiniseg  5367  dmmpt  5368  fmpt  5893  suppss2  6303  suppssfv  6304  suppssov1  6305  cantnfreslem  7634  cantnfres  7636  cantnflem1  7648  cantnf  7652  r0weon  7899  compss  8261  infmsup  9991  eqglact  14996  odngen  15216  rrgsupp  16356  psrbagsn  16560  coe1mul2lem2  16666  pjdm  16939  xkoccn  17656  txcnmpt  17661  txdis1cn  17672  pthaus  17675  txkgen  17689  xkoco1cn  17694  xkoco2cn  17695  xkoinjcn  17724  txcon  17726  imasnopn  17727  imasncld  17728  imasncls  17729  ptcmplem1  18088  ptcmplem3  18090  ptcmplem4  18091  tmdgsum2  18131  symgtgp  18136  tgpconcompeqg  18146  ghmcnp  18149  tgpt0  18153  divstgpopn  18154  divstgphaus  18157  eltsms  18167  prdsxmslem2  18564  efopn  20554  atansopn  20777  xrlimcnp  20812  lgseisenlem3  21140  lgseisenlem4  21141  suppss2f  24054  mbfposadd  26265  cnambfre  26266  itg2addnclem2  26270  iblabsnclem  26281  ftc1anclem1  26293  ftc1anclem6  26298  uvcff  27230  pwfi2f1o  27250
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-ral 2712  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-mpt 4271  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894
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