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Theorem orderseqlem 23420
Description: Lemma for poseq 23421 and soseq 23422. 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 5232 . . . . 5  |-  ( f  =  G  ->  (
f : x --> A  <->  G :
x --> A ) )
21rexbidv 2528 . . . 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 2853 . . 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 5252 . . . . 5  |-  ( G : x --> A  ->  ran  G  C_  A )
7 unss1 3254 . . . . 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 5403 . . . 4  |-  ( G `
 X )  e.  ( ran  G  u.  {
(/) } )
10 ssel 3097 . . . 4  |-  ( ( ran  G  u.  { (/)
} )  C_  ( A  u.  { (/) } )  ->  ( ( G `
 X )  e.  ( ran  G  u.  {
(/) } )  ->  ( G `  X )  e.  ( A  u.  { (/)
} ) ) )
118, 9, 10ee10 1372 . . 3  |-  ( G : x --> A  -> 
( G `  X
)  e.  ( A  u.  { (/) } ) )
1211rexlimivw 2625 . 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 1619    e. wcel 1621   {cab 2239   E.wrex 2510    u. cun 3076    C_ wss 3078   (/)c0 3362   {csn 3544   Oncon0 4285   ran crn 4581   -->wf 4588   ` cfv 4592
This theorem is referenced by:  poseq  23421  soseq  23422
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-fv 4608
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