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Theorem findfvcl 24891
Description: Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
Hypotheses
Ref Expression
findfvcl.1  |-  ( ph  ->  ( F `  (/) )  e.  P )
findfvcl.2  |-  ( y  e.  om  ->  ( ph  ->  ( ( F `
 y )  e.  P  ->  ( F `  suc  y )  e.  P ) ) )
Assertion
Ref Expression
findfvcl  |-  ( A  e.  om  ->  ( ph  ->  ( F `  A )  e.  P
) )
Distinct variable groups:    y, F    y, P    ph, y
Allowed substitution hint:    A( y)

Proof of Theorem findfvcl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveleq 24890 . 2  |-  ( x  =  (/)  ->  ( (
ph  ->  ( F `  x )  e.  P
)  <->  ( ph  ->  ( F `  (/) )  e.  P ) ) )
2 fveleq 24890 . 2  |-  ( x  =  y  ->  (
( ph  ->  ( F `
 x )  e.  P )  <->  ( ph  ->  ( F `  y
)  e.  P ) ) )
3 fveleq 24890 . 2  |-  ( x  =  suc  y  -> 
( ( ph  ->  ( F `  x )  e.  P )  <->  ( ph  ->  ( F `  suc  y )  e.  P
) ) )
4 fveleq 24890 . 2  |-  ( x  =  A  ->  (
( ph  ->  ( F `
 x )  e.  P )  <->  ( ph  ->  ( F `  A
)  e.  P ) ) )
5 findfvcl.1 . 2  |-  ( ph  ->  ( F `  (/) )  e.  P )
6 findfvcl.2 . . 3  |-  ( y  e.  om  ->  ( ph  ->  ( ( F `
 y )  e.  P  ->  ( F `  suc  y )  e.  P ) ) )
76a2d 23 . 2  |-  ( y  e.  om  ->  (
( ph  ->  ( F `
 y )  e.  P )  ->  ( ph  ->  ( F `  suc  y )  e.  P
) ) )
81, 2, 3, 4, 5, 7finds 4682 1  |-  ( A  e.  om  ->  ( ph  ->  ( F `  A )  e.  P
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   (/)c0 3455   suc csuc 4394   omcom 4656   ` cfv 5255
This theorem is referenced by:  findreccl  24892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-iota 5219  df-fv 5263
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