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Theorem elrnmpt1 5251
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 3116 . . . 4  |-  x  e. 
_V
2 id 22 . . . . . . 7  |-  ( x  =  z  ->  x  =  z )
3 csbeq1a 3444 . . . . . . 7  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
42, 3eleq12d 2549 . . . . . 6  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
5 csbeq1a 3444 . . . . . . 7  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
65biantrud 507 . . . . . 6  |-  ( x  =  z  ->  (
z  e.  [_ z  /  x ]_ A  <->  ( z  e.  [_ z  /  x ]_ A  /\  B  = 
[_ z  /  x ]_ B ) ) )
74, 6bitr2d 254 . . . . 5  |-  ( x  =  z  ->  (
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B )  <-> 
x  e.  A ) )
87equcoms 1744 . . . 4  |-  ( z  =  x  ->  (
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B )  <-> 
x  e.  A ) )
91, 8spcev 3205 . . 3  |-  ( x  e.  A  ->  E. z
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) )
10 df-rex 2820 . . . . . 6  |-  ( E. x  e.  A  y  =  B  <->  E. x
( x  e.  A  /\  y  =  B
) )
11 nfv 1683 . . . . . . 7  |-  F/ z ( x  e.  A  /\  y  =  B
)
12 nfcsb1v 3451 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ A
1312nfcri 2622 . . . . . . . 8  |-  F/ x  z  e.  [_ z  /  x ]_ A
14 nfcsb1v 3451 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ B
1514nfeq2 2646 . . . . . . . 8  |-  F/ x  y  =  [_ z  /  x ]_ B
1613, 15nfan 1875 . . . . . . 7  |-  F/ x
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B )
175eqeq2d 2481 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  [_ z  /  x ]_ B ) )
184, 17anbi12d 710 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  A  /\  y  =  B
)  <->  ( z  e. 
[_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) ) )
1911, 16, 18cbvex 1995 . . . . . 6  |-  ( E. x ( x  e.  A  /\  y  =  B )  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) )
2010, 19bitri 249 . . . . 5  |-  ( E. x  e.  A  y  =  B  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) )
21 eqeq1 2471 . . . . . . 7  |-  ( y  =  B  ->  (
y  =  [_ z  /  x ]_ B  <->  B  =  [_ z  /  x ]_ B ) )
2221anbi2d 703 . . . . . 6  |-  ( y  =  B  ->  (
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B )  <-> 
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) ) )
2322exbidv 1690 . . . . 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 257 . . . 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 5248 . . . 4  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
2724, 26elab2g 3252 . . 3  |-  ( B  e.  V  ->  ( B  e.  ran  F  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) ) )
289, 27syl5ibr 221 . 2  |-  ( B  e.  V  ->  (
x  e.  A  ->  B  e.  ran  F ) )
2928impcom 430 1  |-  ( ( x  e.  A  /\  B  e.  V )  ->  B  e.  ran  F
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815   [_csb 3435    |-> cmpt 4505   ran crn 5000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-mpt 4507  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  fliftel1  6196  minveclem4  21610  minvecolem4  25500  rexunirn  27094  totbndbnd  29916  rrnequiv  29962  suprnmpt  31057  fourierdlem31  31466
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