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Theorem orderseqlem 23653
Description: Lemma for poseq 23654 and soseq 23655. The function value of a sequene is either in  A or null. (Contributed by Scott Fenton, 8-Jun-2011.)
Hypothesis
Ref Expression
orderseqlem.1  |-  F  =  { f  |  E. x  e.  On  f : x --> A }
Assertion
Ref Expression
orderseqlem  |-  ( G  e.  F  ->  ( G `  X )  e.  ( A  u.  { (/)
} ) )
Distinct variable groups:    A, f, x    f, G, x    x, X
Allowed substitution hints:    F( x, f)    X( f)

Proof of Theorem orderseqlem
StepHypRef Expression
1 feq1 5340 . . . . 5  |-  ( f  =  G  ->  (
f : x --> A  <->  G :
x --> A ) )
21rexbidv 2565 . . . 4  |-  ( f  =  G  ->  ( E. x  e.  On  f : x --> A  <->  E. x  e.  On  G : x --> A ) )
3 orderseqlem.1 . . . 4  |-  F  =  { f  |  E. x  e.  On  f : x --> A }
42, 3elab2g 2917 . . 3  |-  ( G  e.  F  ->  ( G  e.  F  <->  E. x  e.  On  G : x --> A ) )
54ibi 234 . 2  |-  ( G  e.  F  ->  E. x  e.  On  G : x --> A )
6 frn 5360 . . . . 5  |-  ( G : x --> A  ->  ran  G  C_  A )
7 unss1 3345 . . . . 5  |-  ( ran 
G  C_  A  ->  ( ran  G  u.  { (/)
} )  C_  ( A  u.  { (/) } ) )
86, 7syl 17 . . . 4  |-  ( G : x --> A  -> 
( ran  G  u.  {
(/) } )  C_  ( A  u.  { (/) } ) )
9 fvrn0 5511 . . . 4  |-  ( G `
 X )  e.  ( ran  G  u.  {
(/) } )
10 ssel 3175 . . . 4  |-  ( ( ran  G  u.  { (/)
} )  C_  ( A  u.  { (/) } )  ->  ( ( G `
 X )  e.  ( ran  G  u.  {
(/) } )  ->  ( G `  X )  e.  ( A  u.  { (/)
} ) ) )
118, 9, 10ee10 1368 . . 3  |-  ( G : x --> A  -> 
( G `  X
)  e.  ( A  u.  { (/) } ) )
1211rexlimivw 2664 . 2  |-  ( E. x  e.  On  G : x --> A  -> 
( G `  X
)  e.  ( A  u.  { (/) } ) )
135, 12syl 17 1  |-  ( G  e.  F  ->  ( G `  X )  e.  ( A  u.  { (/)
} ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   {cab 2270   E.wrex 2545    u. cun 3151    C_ wss 3153   (/)c0 3456   {csn 3641   Oncon0 4391   ran crn 4689   -->wf 5217   ` cfv 5221
This theorem is referenced by:  poseq  23654  soseq  23655
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229
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