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Theorem orderseqlem 24323
Description: Lemma for poseq 24324 and soseq 24325. 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 5391 . . . . 5  |-  ( f  =  G  ->  (
f : x --> A  <->  G :
x --> A ) )
21rexbidv 2577 . . . 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 2929 . . 3  |-  ( G  e.  F  ->  ( G  e.  F  <->  E. x  e.  On  G : x --> A ) )
54ibi 232 . 2  |-  ( G  e.  F  ->  E. x  e.  On  G : x --> A )
6 frn 5411 . . . . 5  |-  ( G : x --> A  ->  ran  G  C_  A )
7 unss1 3357 . . . . 5  |-  ( ran 
G  C_  A  ->  ( ran  G  u.  { (/)
} )  C_  ( A  u.  { (/) } ) )
86, 7syl 15 . . . 4  |-  ( G : x --> A  -> 
( ran  G  u.  {
(/) } )  C_  ( A  u.  { (/) } ) )
9 fvrn0 5566 . . . 4  |-  ( G `
 X )  e.  ( ran  G  u.  {
(/) } )
10 ssel 3187 . . . 4  |-  ( ( ran  G  u.  { (/)
} )  C_  ( A  u.  { (/) } )  ->  ( ( G `
 X )  e.  ( ran  G  u.  {
(/) } )  ->  ( G `  X )  e.  ( A  u.  { (/)
} ) ) )
118, 9, 10ee10 1366 . . 3  |-  ( G : x --> A  -> 
( G `  X
)  e.  ( A  u.  { (/) } ) )
1211rexlimivw 2676 . 2  |-  ( E. x  e.  On  G : x --> A  -> 
( G `  X
)  e.  ( A  u.  { (/) } ) )
135, 12syl 15 1  |-  ( G  e.  F  ->  ( G `  X )  e.  ( A  u.  { (/)
} ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   Oncon0 4408   ran crn 4706   -->wf 5267   ` cfv 5271
This theorem is referenced by:  poseq  24324  soseq  24325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
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