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Theorem frsucmptn 6633
Description: The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class 
D is a proper class). This is a technical lemma that can be used together with frsucmpt 6632 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
frsucmpt.1  |-  F/_ x A
frsucmpt.2  |-  F/_ x B
frsucmpt.3  |-  F/_ x D
frsucmpt.4  |-  F  =  ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )
frsucmpt.5  |-  ( x  =  ( F `  B )  ->  C  =  D )
Assertion
Ref Expression
frsucmptn  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )

Proof of Theorem frsucmptn
StepHypRef Expression
1 frsucmpt.4 . . 3  |-  F  =  ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )
21fveq1i 5670 . 2  |-  ( F `
 suc  B )  =  ( ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om ) `  suc  B )
3 frfnom 6629 . . . . . 6  |-  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  Fn  om
4 fndm 5485 . . . . . 6  |-  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )  Fn  om  ->  dom  ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )  =  om )
53, 4ax-mp 8 . . . . 5  |-  dom  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  =  om
65eleq2i 2452 . . . 4  |-  ( suc 
B  e.  dom  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  <->  suc 
B  e.  om )
7 peano2b 4802 . . . . 5  |-  ( B  e.  om  <->  suc  B  e. 
om )
8 frsuc 6631 . . . . . . . 8  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  ( ( x  e.  _V  |->  C ) `  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  B ) ) )
91fveq1i 5670 . . . . . . . . 9  |-  ( F `
 B )  =  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  B )
109fveq2i 5672 . . . . . . . 8  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om ) `  B ) )
118, 10syl6eqr 2438 . . . . . . 7  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) ) )
12 nfmpt1 4240 . . . . . . . . . . . 12  |-  F/_ x
( x  e.  _V  |->  C )
13 frsucmpt.1 . . . . . . . . . . . 12  |-  F/_ x A
1412, 13nfrdg 6609 . . . . . . . . . . 11  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
15 nfcv 2524 . . . . . . . . . . 11  |-  F/_ x om
1614, 15nfres 5089 . . . . . . . . . 10  |-  F/_ x
( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )
171, 16nfcxfr 2521 . . . . . . . . 9  |-  F/_ x F
18 frsucmpt.2 . . . . . . . . 9  |-  F/_ x B
1917, 18nffv 5676 . . . . . . . 8  |-  F/_ x
( F `  B
)
20 frsucmpt.3 . . . . . . . 8  |-  F/_ x D
21 frsucmpt.5 . . . . . . . 8  |-  ( x  =  ( F `  B )  ->  C  =  D )
22 eqid 2388 . . . . . . . 8  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
2319, 20, 21, 22fvmptnf 5762 . . . . . . 7  |-  ( -.  D  e.  _V  ->  ( ( x  e.  _V  |->  C ) `  ( F `  B )
)  =  (/) )
2411, 23sylan9eqr 2442 . . . . . 6  |-  ( ( -.  D  e.  _V  /\  B  e.  om )  ->  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  suc  B )  =  (/) )
2524ex 424 . . . . 5  |-  ( -.  D  e.  _V  ->  ( B  e.  om  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  (/) ) )
267, 25syl5bir 210 . . . 4  |-  ( -.  D  e.  _V  ->  ( suc  B  e.  om  ->  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  suc  B )  =  (/) ) )
276, 26syl5bi 209 . . 3  |-  ( -.  D  e.  _V  ->  ( suc  B  e.  dom  ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B
)  =  (/) ) )
28 ndmfv 5696 . . 3  |-  ( -. 
suc  B  e.  dom  ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B
)  =  (/) )
2927, 28pm2.61d1 153 . 2  |-  ( -.  D  e.  _V  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  (/) )
302, 29syl5eq 2432 1  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717   F/_wnfc 2511   _Vcvv 2900   (/)c0 3572    e. cmpt 4208   suc csuc 4525   omcom 4786   dom cdm 4819    |` cres 4821    Fn wfn 5390   ` cfv 5395   reccrdg 6604
This theorem is referenced by:  trpredlem1  25255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-recs 6570  df-rdg 6605
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