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Theorem findfvcl 26204
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 26203 . 2  |-  ( x  =  (/)  ->  ( (
ph  ->  ( F `  x )  e.  P
)  <->  ( ph  ->  ( F `  (/) )  e.  P ) ) )
2 fveleq 26203 . 2  |-  ( x  =  y  ->  (
( ph  ->  ( F `
 x )  e.  P )  <->  ( ph  ->  ( F `  y
)  e.  P ) ) )
3 fveleq 26203 . 2  |-  ( x  =  suc  y  -> 
( ( ph  ->  ( F `  x )  e.  P )  <->  ( ph  ->  ( F `  suc  y )  e.  P
) ) )
4 fveleq 26203 . 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 25 . 2  |-  ( y  e.  om  ->  (
( ph  ->  ( F `
 y )  e.  P )  ->  ( ph  ->  ( F `  suc  y )  e.  P
) ) )
81, 2, 3, 4, 5, 7finds 4873 1  |-  ( A  e.  om  ->  ( ph  ->  ( F `  A )  e.  P
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   (/)c0 3630   suc csuc 4585   omcom 4847   ` cfv 5456
This theorem is referenced by:  findreccl  26205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-iota 5420  df-fv 5464
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