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Theorem orderseqlem 25532
Description: Lemma for poseq 25533 and soseq 25534. 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 5579 . . . . 5  |-  ( f  =  G  ->  (
f : x --> A  <->  G :
x --> A ) )
21rexbidv 2728 . . . 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 3086 . . 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 5600 . . . . 5  |-  ( G : x --> A  ->  ran  G  C_  A )
7 unss1 3518 . . . . 5  |-  ( ran 
G  C_  A  ->  ( ran  G  u.  { (/)
} )  C_  ( A  u.  { (/) } ) )
86, 7syl 16 . . . 4  |-  ( G : x --> A  -> 
( ran  G  u.  {
(/) } )  C_  ( A  u.  { (/) } ) )
9 fvrn0 5756 . . . 4  |-  ( G `
 X )  e.  ( ran  G  u.  {
(/) } )
10 ssel 3344 . . . 4  |-  ( ( ran  G  u.  { (/)
} )  C_  ( A  u.  { (/) } )  ->  ( ( G `
 X )  e.  ( ran  G  u.  {
(/) } )  ->  ( G `  X )  e.  ( A  u.  { (/)
} ) ) )
118, 9, 10ee10 1386 . . 3  |-  ( G : x --> A  -> 
( G `  X
)  e.  ( A  u.  { (/) } ) )
1211rexlimivw 2828 . 2  |-  ( E. x  e.  On  G : x --> A  -> 
( G `  X
)  e.  ( A  u.  { (/) } ) )
135, 12syl 16 1  |-  ( G  e.  F  ->  ( G `  X )  e.  ( A  u.  { (/)
} ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {cab 2424   E.wrex 2708    u. cun 3320    C_ wss 3322   (/)c0 3630   {csn 3816   Oncon0 4584   ran crn 4882   -->wf 5453   ` cfv 5457
This theorem is referenced by:  poseq  25533  soseq  25534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
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-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  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-uni 4018  df-br 4216  df-opab 4270  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465
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