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Theorem frsucmptn 6688
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 6687 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 5721 . 2  |-  ( F `
 suc  B )  =  ( ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om ) `  suc  B )
3 frfnom 6684 . . . . . 6  |-  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  Fn  om
4 fndm 5536 . . . . . 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 2499 . . . 4  |-  ( suc 
B  e.  dom  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  <->  suc 
B  e.  om )
7 peano2b 4853 . . . . 5  |-  ( B  e.  om  <->  suc  B  e. 
om )
8 frsuc 6686 . . . . . . . 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 5721 . . . . . . . . 9  |-  ( F `
 B )  =  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  B )
109fveq2i 5723 . . . . . . . 8  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om ) `  B ) )
118, 10syl6eqr 2485 . . . . . . 7  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) ) )
12 nfmpt1 4290 . . . . . . . . . . . 12  |-  F/_ x
( x  e.  _V  |->  C )
13 frsucmpt.1 . . . . . . . . . . . 12  |-  F/_ x A
1412, 13nfrdg 6664 . . . . . . . . . . 11  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
15 nfcv 2571 . . . . . . . . . . 11  |-  F/_ x om
1614, 15nfres 5140 . . . . . . . . . 10  |-  F/_ x
( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )
171, 16nfcxfr 2568 . . . . . . . . 9  |-  F/_ x F
18 frsucmpt.2 . . . . . . . . 9  |-  F/_ x B
1917, 18nffv 5727 . . . . . . . 8  |-  F/_ x
( F `  B
)
20 frsucmpt.3 . . . . . . . 8  |-  F/_ x D
21 frsucmpt.5 . . . . . . . 8  |-  ( x  =  ( F `  B )  ->  C  =  D )
22 eqid 2435 . . . . . . . 8  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
2319, 20, 21, 22fvmptnf 5814 . . . . . . 7  |-  ( -.  D  e.  _V  ->  ( ( x  e.  _V  |->  C ) `  ( F `  B )
)  =  (/) )
2411, 23sylan9eqr 2489 . . . . . 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 5747 . . 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 2479 1  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725   F/_wnfc 2558   _Vcvv 2948   (/)c0 3620    e. cmpt 4258   suc csuc 4575   omcom 4837   dom cdm 4870    |` cres 4872    Fn wfn 5441   ` cfv 5446   reccrdg 6659
This theorem is referenced by:  trpredlem1  25497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660
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