Users' Mathboxes Mathbox for Jeff Hoffman < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  findabrcl Unicode version

Theorem findabrcl 25452
Description: Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
findabrcl.1  |-  ( z  e.  P  ->  ( G `  z )  e.  P )
Assertion
Ref Expression
findabrcl  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( ( x  e. 
_V  |->  ( rec ( G ,  A ) `  x ) ) `  C )  e.  P
)
Distinct variable groups:    x, G    x, A    x, C    z, G    z, A    z, P
Allowed substitution hints:    C( z)    P( x)

Proof of Theorem findabrcl
StepHypRef Expression
1 elex 2872 . . . 4  |-  ( C  e.  om  ->  C  e.  _V )
2 fveq2 5605 . . . . 5  |-  ( x  =  C  ->  ( rec ( G ,  A
) `  x )  =  ( rec ( G ,  A ) `  C ) )
3 eqid 2358 . . . . 5  |-  ( x  e.  _V  |->  ( rec ( G ,  A
) `  x )
)  =  ( x  e.  _V  |->  ( rec ( G ,  A
) `  x )
)
4 fvex 5619 . . . . 5  |-  ( rec ( G ,  A
) `  C )  e.  _V
52, 3, 4fvmpt 5682 . . . 4  |-  ( C  e.  _V  ->  (
( x  e.  _V  |->  ( rec ( G ,  A ) `  x
) ) `  C
)  =  ( rec ( G ,  A
) `  C )
)
61, 5syl 15 . . 3  |-  ( C  e.  om  ->  (
( x  e.  _V  |->  ( rec ( G ,  A ) `  x
) ) `  C
)  =  ( rec ( G ,  A
) `  C )
)
76adantr 451 . 2  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( ( x  e. 
_V  |->  ( rec ( G ,  A ) `  x ) ) `  C )  =  ( rec ( G ,  A ) `  C
) )
8 findabrcl.1 . . . 4  |-  ( z  e.  P  ->  ( G `  z )  e.  P )
98findreccl 25451 . . 3  |-  ( C  e.  om  ->  ( A  e.  P  ->  ( rec ( G ,  A ) `  C
)  e.  P ) )
109imp 418 . 2  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( rec ( G ,  A ) `  C )  e.  P
)
117, 10eqeltrd 2432 1  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( ( x  e. 
_V  |->  ( rec ( G ,  A ) `  x ) ) `  C )  e.  P
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864    e. cmpt 4156   omcom 4735   ` cfv 5334   reccrdg 6506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-recs 6472  df-rdg 6507
  Copyright terms: Public domain W3C validator