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Theorem elrnmpt1 4613
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 2577 . . . 4  |-  x  e. 
_V
2 id 19 . . . . . . 7  |-  ( x  =  z  ->  x  =  z )
3 csbeq1a 2888 . . . . . . 7  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
42, 3eleq12d 2124 . . . . . 6  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
5 csbeq1a 2888 . . . . . . 7  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
65biantrud 292 . . . . . 6  |-  ( x  =  z  ->  (
z  e.  [_ z  /  x ]_ A  <->  ( z  e.  [_ z  /  x ]_ A  /\  B  = 
[_ z  /  x ]_ B ) ) )
74, 6bitr2d 182 . . . . 5  |-  ( x  =  z  ->  (
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B )  <-> 
x  e.  A ) )
87equcoms 1610 . . . 4  |-  ( z  =  x  ->  (
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B )  <-> 
x  e.  A ) )
91, 8spcev 2664 . . 3  |-  ( x  e.  A  ->  E. z
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) )
10 df-rex 2329 . . . . . 6  |-  ( E. x  e.  A  y  =  B  <->  E. x
( x  e.  A  /\  y  =  B
) )
11 nfv 1437 . . . . . . 7  |-  F/ z ( x  e.  A  /\  y  =  B
)
12 nfcsb1v 2910 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ A
1312nfcri 2188 . . . . . . . 8  |-  F/ x  z  e.  [_ z  /  x ]_ A
14 nfcsb1v 2910 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ B
1514nfeq2 2205 . . . . . . . 8  |-  F/ x  y  =  [_ z  /  x ]_ B
1613, 15nfan 1473 . . . . . . 7  |-  F/ x
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B )
175eqeq2d 2067 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  [_ z  /  x ]_ B ) )
184, 17anbi12d 450 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  A  /\  y  =  B
)  <->  ( z  e. 
[_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) ) )
1911, 16, 18cbvex 1655 . . . . . 6  |-  ( E. x ( x  e.  A  /\  y  =  B )  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) )
2010, 19bitri 177 . . . . 5  |-  ( E. x  e.  A  y  =  B  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) )
21 eqeq1 2062 . . . . . . 7  |-  ( y  =  B  ->  (
y  =  [_ z  /  x ]_ B  <->  B  =  [_ z  /  x ]_ B ) )
2221anbi2d 445 . . . . . 6  |-  ( y  =  B  ->  (
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B )  <-> 
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) ) )
2322exbidv 1722 . . . . 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 185 . . . 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 4610 . . . 4  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
2724, 26elab2g 2712 . . 3  |-  ( B  e.  V  ->  ( B  e.  ran  F  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) ) )
289, 27syl5ibr 149 . 2  |-  ( B  e.  V  ->  (
x  e.  A  ->  B  e.  ran  F ) )
2928impcom 120 1  |-  ( ( x  e.  A  /\  B  e.  V )  ->  B  e.  ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324   [_csb 2880    |-> cmpt 3846   ran crn 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-mpt 3848  df-cnv 4381  df-dm 4383  df-rn 4384
This theorem is referenced by:  fliftel1  5462
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