MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frsucmptn Structured version   Visualization version   Unicode version

Theorem frsucmptn 7174
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 7173 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 5880 . 2  |-  ( F `
 suc  B )  =  ( ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om ) `  suc  B )
3 frfnom 7170 . . . . . 6  |-  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  Fn  om
4 fndm 5685 . . . . . 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 2541 . . . 4  |-  ( suc 
B  e.  dom  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  <->  suc 
B  e.  om )
7 peano2b 6727 . . . . 5  |-  ( B  e.  om  <->  suc  B  e. 
om )
8 frsuc 7172 . . . . . . . 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 5880 . . . . . . . . 9  |-  ( F `
 B )  =  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  B )
109fveq2i 5882 . . . . . . . 8  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om ) `  B ) )
118, 10syl6eqr 2523 . . . . . . 7  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) ) )
12 nfmpt1 4485 . . . . . . . . . . . 12  |-  F/_ x
( x  e.  _V  |->  C )
13 frsucmpt.1 . . . . . . . . . . . 12  |-  F/_ x A
1412, 13nfrdg 7150 . . . . . . . . . . 11  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
15 nfcv 2612 . . . . . . . . . . 11  |-  F/_ x om
1614, 15nfres 5113 . . . . . . . . . 10  |-  F/_ x
( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )
171, 16nfcxfr 2610 . . . . . . . . 9  |-  F/_ x F
18 frsucmpt.2 . . . . . . . . 9  |-  F/_ x B
1917, 18nffv 5886 . . . . . . . 8  |-  F/_ x
( F `  B
)
20 frsucmpt.3 . . . . . . . 8  |-  F/_ x D
21 frsucmpt.5 . . . . . . . 8  |-  ( x  =  ( F `  B )  ->  C  =  D )
22 eqid 2471 . . . . . . . 8  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
2319, 20, 21, 22fvmptnf 5982 . . . . . . 7  |-  ( -.  D  e.  _V  ->  ( ( x  e.  _V  |->  C ) `  ( F `  B )
)  =  (/) )
2411, 23sylan9eqr 2527 . . . . . 6  |-  ( ( -.  D  e.  _V  /\  B  e.  om )  ->  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  suc  B )  =  (/) )
2524ex 441 . . . . 5  |-  ( -.  D  e.  _V  ->  ( B  e.  om  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  (/) ) )
267, 25syl5bir 226 . . . 4  |-  ( -.  D  e.  _V  ->  ( suc  B  e.  om  ->  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  suc  B )  =  (/) ) )
276, 26syl5bi 225 . . 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 5903 . . 3  |-  ( -. 
suc  B  e.  dom  ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B
)  =  (/) )
2927, 28pm2.61d1 164 . 2  |-  ( -.  D  e.  _V  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  (/) )
302, 29syl5eq 2517 1  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1452    e. wcel 1904   F/_wnfc 2599   _Vcvv 3031   (/)c0 3722    |-> cmpt 4454   dom cdm 4839    |` cres 4841   suc csuc 5432    Fn wfn 5584   ` cfv 5589   omcom 6711   reccrdg 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146
This theorem is referenced by:  trpredlem1  30539
  Copyright terms: Public domain W3C validator