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Theorem findfvcl 24963
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 24962 . 2  |-  ( x  =  (/)  ->  ( (
ph  ->  ( F `  x )  e.  P
)  <->  ( ph  ->  ( F `  (/) )  e.  P ) ) )
2 fveleq 24962 . 2  |-  ( x  =  y  ->  (
( ph  ->  ( F `
 x )  e.  P )  <->  ( ph  ->  ( F `  y
)  e.  P ) ) )
3 fveleq 24962 . 2  |-  ( x  =  suc  y  -> 
( ( ph  ->  ( F `  x )  e.  P )  <->  ( ph  ->  ( F `  suc  y )  e.  P
) ) )
4 fveleq 24962 . 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 4698 1  |-  ( A  e.  om  ->  ( ph  ->  ( F `  A )  e.  P
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   (/)c0 3468   suc csuc 4410   omcom 4672   ` cfv 5271
This theorem is referenced by:  findreccl  24964
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-pr 4230  ax-un 4528
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 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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-iota 5235  df-fv 5279
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