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Theorem frsucmptn 6467
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 6466 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 5542 . 2  |-  ( F `
 suc  B )  =  ( ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om ) `  suc  B )
3 frfnom 6463 . . . . . 6  |-  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  Fn  om
4 fndm 5359 . . . . . 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 2360 . . . 4  |-  ( suc 
B  e.  dom  ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om )  <->  suc 
B  e.  om )
7 peano2b 4688 . . . . 5  |-  ( B  e.  om  <->  suc  B  e. 
om )
8 frsuc 6465 . . . . . . . 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 5542 . . . . . . . . 9  |-  ( F `
 B )  =  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  B )
109fveq2i 5544 . . . . . . . 8  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( ( rec ( ( x  e. 
_V  |->  C ) ,  A )  |`  om ) `  B ) )
118, 10syl6eqr 2346 . . . . . . 7  |-  ( B  e.  om  ->  (
( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) ) )
12 nfmpt1 4125 . . . . . . . . . . . 12  |-  F/_ x
( x  e.  _V  |->  C )
13 frsucmpt.1 . . . . . . . . . . . 12  |-  F/_ x A
1412, 13nfrdg 6443 . . . . . . . . . . 11  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
15 nfcv 2432 . . . . . . . . . . 11  |-  F/_ x om
1614, 15nfres 4973 . . . . . . . . . 10  |-  F/_ x
( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )
171, 16nfcxfr 2429 . . . . . . . . 9  |-  F/_ x F
18 frsucmpt.2 . . . . . . . . 9  |-  F/_ x B
1917, 18nffv 5548 . . . . . . . 8  |-  F/_ x
( F `  B
)
20 frsucmpt.3 . . . . . . . 8  |-  F/_ x D
21 frsucmpt.5 . . . . . . . 8  |-  ( x  =  ( F `  B )  ->  C  =  D )
22 eqid 2296 . . . . . . . 8  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
2319, 20, 21, 22fvmptnf 5633 . . . . . . 7  |-  ( -.  D  e.  _V  ->  ( ( x  e.  _V  |->  C ) `  ( F `  B )
)  =  (/) )
2411, 23sylan9eqr 2350 . . . . . 6  |-  ( ( -.  D  e.  _V  /\  B  e.  om )  ->  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  suc  B )  =  (/) )
2524ex 423 . . . . 5  |-  ( -.  D  e.  _V  ->  ( B  e.  om  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  (/) ) )
267, 25syl5bir 209 . . . 4  |-  ( -.  D  e.  _V  ->  ( suc  B  e.  om  ->  ( ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om ) `  suc  B )  =  (/) ) )
276, 26syl5bi 208 . . 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 5568 . . 3  |-  ( -. 
suc  B  e.  dom  ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B
)  =  (/) )
2927, 28pm2.61d1 151 . 2  |-  ( -.  D  e.  _V  ->  ( ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om ) `  suc  B )  =  (/) )
302, 29syl5eq 2340 1  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   F/_wnfc 2419   _Vcvv 2801   (/)c0 3468    e. cmpt 4093   suc csuc 4410   omcom 4672   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  trpredlem1  24301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
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