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Theorem orderseqlem 25507
Description: Lemma for poseq 25508 and soseq 25509. 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 5567 . . . . 5  |-  ( f  =  G  ->  (
f : x --> A  <->  G :
x --> A ) )
21rexbidv 2718 . . . 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 3076 . . 3  |-  ( G  e.  F  ->  ( G  e.  F  <->  E. x  e.  On  G : x --> A ) )
54ibi 233 . 2  |-  ( G  e.  F  ->  E. x  e.  On  G : x --> A )
6 frn 5588 . . . . 5  |-  ( G : x --> A  ->  ran  G  C_  A )
7 unss1 3508 . . . . 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 5744 . . . 4  |-  ( G `
 X )  e.  ( ran  G  u.  {
(/) } )
10 ssel 3334 . . . 4  |-  ( ( ran  G  u.  { (/)
} )  C_  ( A  u.  { (/) } )  ->  ( ( G `
 X )  e.  ( ran  G  u.  {
(/) } )  ->  ( G `  X )  e.  ( A  u.  { (/)
} ) ) )
118, 9, 10ee10 1385 . . 3  |-  ( G : x --> A  -> 
( G `  X
)  e.  ( A  u.  { (/) } ) )
1211rexlimivw 2818 . 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 1652    e. wcel 1725   {cab 2421   E.wrex 2698    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   Oncon0 4573   ran crn 4870   -->wf 5441   ` cfv 5445
This theorem is referenced by:  poseq  25508  soseq  25509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-fv 5453
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