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Theorem frsucmptn 7096
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 7095 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 5849 . 2  |-  ( F `
 suc  B )  =  ( ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om ) `  suc  B )
3 frfnom 7092 . . . . . 6  |-  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  Fn  om
4 fndm 5662 . . . . . 6  |-  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )  Fn  om  ->  dom  ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )  =  om )
53, 4ax-mp 5 . . . . 5  |-  dom  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  =  om
65eleq2i 2532 . . . 4  |-  ( suc 
B  e.  dom  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  <->  suc 
B  e.  om )
7 peano2b 6689 . . . . 5  |-  ( B  e.  om  <->  suc  B  e. 
om )
8 frsuc 7094 . . . . . . . 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 5849 . . . . . . . . 9  |-  ( F `
 B )  =  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  B )
109fveq2i 5851 . . . . . . . 8  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om ) `  B ) )
118, 10syl6eqr 2513 . . . . . . 7  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) ) )
12 nfmpt1 4528 . . . . . . . . . . . 12  |-  F/_ x
( x  e.  _V  |->  C )
13 frsucmpt.1 . . . . . . . . . . . 12  |-  F/_ x A
1412, 13nfrdg 7072 . . . . . . . . . . 11  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
15 nfcv 2616 . . . . . . . . . . 11  |-  F/_ x om
1614, 15nfres 5264 . . . . . . . . . 10  |-  F/_ x
( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )
171, 16nfcxfr 2614 . . . . . . . . 9  |-  F/_ x F
18 frsucmpt.2 . . . . . . . . 9  |-  F/_ x B
1917, 18nffv 5855 . . . . . . . 8  |-  F/_ x
( F `  B
)
20 frsucmpt.3 . . . . . . . 8  |-  F/_ x D
21 frsucmpt.5 . . . . . . . 8  |-  ( x  =  ( F `  B )  ->  C  =  D )
22 eqid 2454 . . . . . . . 8  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
2319, 20, 21, 22fvmptnf 5949 . . . . . . 7  |-  ( -.  D  e.  _V  ->  ( ( x  e.  _V  |->  C ) `  ( F `  B )
)  =  (/) )
2411, 23sylan9eqr 2517 . . . . . 6  |-  ( ( -.  D  e.  _V  /\  B  e.  om )  ->  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  suc  B )  =  (/) )
2524ex 432 . . . . 5  |-  ( -.  D  e.  _V  ->  ( B  e.  om  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  (/) ) )
267, 25syl5bir 218 . . . 4  |-  ( -.  D  e.  _V  ->  ( suc  B  e.  om  ->  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  suc  B )  =  (/) ) )
276, 26syl5bi 217 . . 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 5872 . . 3  |-  ( -. 
suc  B  e.  dom  ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B
)  =  (/) )
2927, 28pm2.61d1 159 . 2  |-  ( -.  D  e.  _V  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  (/) )
302, 29syl5eq 2507 1  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1398    e. wcel 1823   F/_wnfc 2602   _Vcvv 3106   (/)c0 3783    |-> cmpt 4497   suc csuc 4869   dom cdm 4988    |` cres 4990    Fn wfn 5565   ` cfv 5570   omcom 6673   reccrdg 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068
This theorem is referenced by:  trpredlem1  29550
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