MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elrnmpt1 Structured version   Unicode version

Theorem elrnmpt1 5121
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 2961 . . . 4  |-  x  e. 
_V
2 id 21 . . . . . . 7  |-  ( x  =  z  ->  x  =  z )
3 csbeq1a 3261 . . . . . . 7  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
42, 3eleq12d 2506 . . . . . 6  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
5 csbeq1a 3261 . . . . . . 7  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
65biantrud 495 . . . . . 6  |-  ( x  =  z  ->  (
z  e.  [_ z  /  x ]_ A  <->  ( z  e.  [_ z  /  x ]_ A  /\  B  = 
[_ z  /  x ]_ B ) ) )
74, 6bitr2d 247 . . . . 5  |-  ( x  =  z  ->  (
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B )  <-> 
x  e.  A ) )
87equcoms 1694 . . . 4  |-  ( z  =  x  ->  (
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B )  <-> 
x  e.  A ) )
91, 8spcev 3045 . . 3  |-  ( x  e.  A  ->  E. z
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) )
10 df-rex 2713 . . . . . 6  |-  ( E. x  e.  A  y  =  B  <->  E. x
( x  e.  A  /\  y  =  B
) )
11 nfv 1630 . . . . . . 7  |-  F/ z ( x  e.  A  /\  y  =  B
)
12 nfcsb1v 3285 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ A
1312nfcri 2568 . . . . . . . 8  |-  F/ x  z  e.  [_ z  /  x ]_ A
14 nfcsb1v 3285 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ B
1514nfeq2 2585 . . . . . . . 8  |-  F/ x  y  =  [_ z  /  x ]_ B
1613, 15nfan 1847 . . . . . . 7  |-  F/ x
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B )
175eqeq2d 2449 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  [_ z  /  x ]_ B ) )
184, 17anbi12d 693 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  A  /\  y  =  B
)  <->  ( z  e. 
[_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) ) )
1911, 16, 18cbvex 1984 . . . . . 6  |-  ( E. x ( x  e.  A  /\  y  =  B )  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) )
2010, 19bitri 242 . . . . 5  |-  ( E. x  e.  A  y  =  B  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B ) )
21 eqeq1 2444 . . . . . . 7  |-  ( y  =  B  ->  (
y  =  [_ z  /  x ]_ B  <->  B  =  [_ z  /  x ]_ B ) )
2221anbi2d 686 . . . . . 6  |-  ( y  =  B  ->  (
( z  e.  [_ z  /  x ]_ A  /\  y  =  [_ z  /  x ]_ B )  <-> 
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) ) )
2322exbidv 1637 . . . . 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 250 . . . 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 5118 . . . 4  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
2724, 26elab2g 3086 . . 3  |-  ( B  e.  V  ->  ( B  e.  ran  F  <->  E. z
( z  e.  [_ z  /  x ]_ A  /\  B  =  [_ z  /  x ]_ B ) ) )
289, 27syl5ibr 214 . 2  |-  ( B  e.  V  ->  (
x  e.  A  ->  B  e.  ran  F ) )
2928impcom 421 1  |-  ( ( x  e.  A  /\  B  e.  V )  ->  B  e.  ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   E.wrex 2708   [_csb 3253    e. cmpt 4268   ran crn 4881
This theorem is referenced by:  fliftel1  6034  minveclem4  19335  minvecolem4  22384  rexunirn  23980  totbndbnd  26500  rrnequiv  26546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pr 4405
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-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  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 4215  df-opab 4269  df-mpt 4270  df-cnv 4888  df-dm 4890  df-rn 4891
  Copyright terms: Public domain W3C validator