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Theorem mptpreima 5182
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 4095 . . . . . 6  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
31, 2eqtri 2316 . . . . 5  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
43cnveqi 4872 . . . 4  |-  `' F  =  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
5 cnvopab 5099 . . . 4  |-  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }
64, 5eqtri 2316 . . 3  |-  `' F  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }
76imaeq1i 5025 . 2  |-  ( `' F " C )  =  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) } " C )
8 df-ima 4718 . . 3  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) } " C
)  =  ran  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  |`  C )
9 resopab 5012 . . . . 5  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) }  |`  C )  =  { <. y ,  x >.  |  (
y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }
109rneqi 4921 . . . 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 1572 . . . . . . 7  |-  ( E. y ( y  e.  C  /\  ( x  e.  A  /\  y  =  B ) )  <->  E. y
( x  e.  A  /\  ( y  =  B  /\  y  e.  C
) ) )
15 19.42v 1858 . . . . . . . 8  |-  ( E. y ( x  e.  A  /\  ( y  =  B  /\  y  e.  C ) )  <->  ( x  e.  A  /\  E. y
( y  =  B  /\  y  e.  C
) ) )
16 df-clel 2292 . . . . . . . . . 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 2408 . . . . 5  |-  { x  |  E. y ( y  e.  C  /\  (
x  e.  A  /\  y  =  B )
) }  =  {
x  |  ( x  e.  A  /\  B  e.  C ) }
22 rnopab 4940 . . . . 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 2565 . . . . 5  |-  { x  e.  A  |  B  e.  C }  =  {
x  |  ( x  e.  A  /\  B  e.  C ) }
2421, 22, 233eqtr4i 2326 . . . 4  |-  ran  { <. y ,  x >.  |  ( y  e.  C  /\  ( x  e.  A  /\  y  =  B
) ) }  =  { x  e.  A  |  B  e.  C }
2510, 24eqtri 2316 . . 3  |-  ran  ( { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  |`  C )  =  { x  e.  A  |  B  e.  C }
268, 25eqtri 2316 . 2  |-  ( {
<. y ,  x >.  |  ( x  e.  A  /\  y  =  B
) } " C
)  =  { x  e.  A  |  B  e.  C }
277, 26eqtri 2316 1  |-  ( `' F " C )  =  { x  e.  A  |  B  e.  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560   {copab 4092    e. cmpt 4093   `'ccnv 4704   ran crn 4706    |` cres 4707   "cima 4708
This theorem is referenced by:  mptiniseg  5183  dmmpt  5184  fmpt  5697  suppss2  6089  suppssfv  6090  suppssov1  6091  cantnfreslem  7393  cantnfres  7395  cantnflem1  7407  cantnf  7411  r0weon  7656  compss  8018  infmsup  9748  eqglact  14684  odngen  14904  rrgsupp  16048  psrbagsn  16252  coe1mul2lem2  16361  pjdm  16623  xkoccn  17329  txcnmpt  17334  txdis1cn  17345  pthaus  17348  txkgen  17362  xkoco1cn  17367  xkoco2cn  17368  xkoinjcn  17397  txcon  17399  ptcmplem1  17762  ptcmplem3  17764  ptcmplem4  17765  tmdgsum2  17795  symgtgp  17800  tgpconcompeqg  17810  ghmcnp  17813  tgpt0  17817  divstgpopn  17818  divstgphaus  17821  eltsms  17831  prdsxmslem2  18091  efopn  20021  atansopn  20244  xrlimcnp  20279  lgseisenlem3  20606  lgseisenlem4  20607  suppss2f  23216  itg2addnclem2  25004  itgaddnclem2  25010  iblabsnclem  25014  uvcff  27343  pwfi2f1o  27363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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