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Theorem elrnmpt1 4758
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rnmpt.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
elrnmpt1  |-  ( ( x  e.  A  /\  B  e.  V )  ->  B  e.  ran  F
)

Proof of Theorem elrnmpt1
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2661 . . . 4  |-  x  e. 
_V
2 id 19 . . . . . . 7  |-  ( x  =  z  ->  x  =  z )
3 csbeq1a 2981 . . . . . . 7  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
42, 3eleq12d 2186 . . . . . 6  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
5 csbeq1a 2981 . . . . . . 7  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
65biantrud 300 . . . . . 6  |-  ( x  =  z  ->  (
z  e.  [_ z  /  x ]_ A  <->  ( z  e.  [_ z  /  x ]_ A  /\  B  = 
[_ z  /  x ]_ B ) ) )
74, 6bitr2d 188 . . . . 5  |-  ( x  =  z  ->  (
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B )  <-> 
x  e.  A ) )
87equcoms 1667 . . . 4  |-  ( z  =  x  ->  (
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B )  <-> 
x  e.  A ) )
91, 8spcev 2752 . . 3  |-  ( x  e.  A  ->  E. z
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) )
10 df-rex 2397 . . . . . 6  |-  ( E. x  e.  A  y  =  B  <->  E. x
( x  e.  A  /\  y  =  B
) )
11 nfv 1491 . . . . . . 7  |-  F/ z ( x  e.  A  /\  y  =  B
)
12 nfcsb1v 3003 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ A
1312nfcri 2250 . . . . . . . 8  |-  F/ x  z  e.  [_ z  /  x ]_ A
14 nfcsb1v 3003 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ B
1514nfeq2 2268 . . . . . . . 8  |-  F/ x  y  =  [_ z  /  x ]_ B
1613, 15nfan 1527 . . . . . . 7  |-  F/ x
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B )
175eqeq2d 2127 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  [_ z  /  x ]_ B ) )
184, 17anbi12d 462 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  A  /\  y  =  B
)  <->  ( z  e. 
[_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) ) )
1911, 16, 18cbvex 1712 . . . . . 6  |-  ( E. x ( x  e.  A  /\  y  =  B )  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) )
2010, 19bitri 183 . . . . 5  |-  ( E. x  e.  A  y  =  B  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) )
21 eqeq1 2122 . . . . . . 7  |-  ( y  =  B  ->  (
y  =  [_ z  /  x ]_ B  <->  B  =  [_ z  /  x ]_ B ) )
2221anbi2d 457 . . . . . 6  |-  ( y  =  B  ->  (
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B )  <-> 
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) ) )
2322exbidv 1779 . . . . 5  |-  ( y  =  B  ->  ( E. z ( z  e. 
[_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B )  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) ) )
2420, 23syl5bb 191 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  B  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) ) )
25 rnmpt.1 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
2625rnmpt 4755 . . . 4  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
2724, 26elab2g 2802 . . 3  |-  ( B  e.  V  ->  ( B  e.  ran  F  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) ) )
289, 27syl5ibr 155 . 2  |-  ( B  e.  V  ->  (
x  e.  A  ->  B  e.  ran  F ) )
2928impcom 124 1  |-  ( ( x  e.  A  /\  B  e.  V )  ->  B  e.  ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   E.wrex 2392   [_csb 2973    |-> cmpt 3957   ran crn 4508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-mpt 3959  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  fliftel1  5661
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