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Theorem rdgivallem 6246
Description: Value of the recursive definition generator. Lemma for rdgival 6247 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)
Assertion
Ref Expression
rdgivallem  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  ( rec ( F ,  A ) `  B )  =  ( A  u.  U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `  x
) ) ) )
Distinct variable groups:    x, A    x, B    x, F    x, V

Proof of Theorem rdgivallem
Dummy variables  g  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-irdg 6235 . . . 4  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
2 rdgruledefgg 6240 . . . . 5  |-  ( ( F  Fn  _V  /\  A  e.  V )  ->  ( Fun  ( g  e.  _V  |->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) ) )  /\  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  y )  e.  _V ) )
32alrimiv 1830 . . . 4  |-  ( ( F  Fn  _V  /\  A  e.  V )  ->  A. y ( Fun  ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) )  /\  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  y )  e.  _V ) )
41, 3tfri2d 6201 . . 3  |-  ( ( ( F  Fn  _V  /\  A  e.  V )  /\  B  e.  On )  ->  ( rec ( F ,  A ) `  B )  =  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  ( rec ( F ,  A
)  |`  B ) ) )
543impa 1161 . 2  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  ( rec ( F ,  A ) `  B )  =  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  ( rec ( F ,  A
)  |`  B ) ) )
6 eqidd 2118 . . 3  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) )  =  ( g  e.  _V  |->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) ) ) )
7 dmeq 4709 . . . . . 6  |-  ( g  =  ( rec ( F ,  A )  |`  B )  ->  dom  g  =  dom  ( rec ( F ,  A
)  |`  B ) )
8 onss 4379 . . . . . . . . 9  |-  ( B  e.  On  ->  B  C_  On )
983ad2ant3 989 . . . . . . . 8  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  B  C_  On )
10 rdgifnon 6244 . . . . . . . . . 10  |-  ( ( F  Fn  _V  /\  A  e.  V )  ->  rec ( F ,  A )  Fn  On )
11 fndm 5192 . . . . . . . . . 10  |-  ( rec ( F ,  A
)  Fn  On  ->  dom 
rec ( F ,  A )  =  On )
1210, 11syl 14 . . . . . . . . 9  |-  ( ( F  Fn  _V  /\  A  e.  V )  ->  dom  rec ( F ,  A )  =  On )
13123adant3 986 . . . . . . . 8  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  dom  rec ( F ,  A )  =  On )
149, 13sseqtrrd 3106 . . . . . . 7  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  B  C_  dom  rec ( F ,  A )
)
15 ssdmres 4811 . . . . . . 7  |-  ( B 
C_  dom  rec ( F ,  A )  <->  dom  ( rec ( F ,  A )  |`  B )  =  B )
1614, 15sylib 121 . . . . . 6  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  dom  ( rec ( F ,  A )  |`  B )  =  B )
177, 16sylan9eqr 2172 . . . . 5  |-  ( ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  /\  g  =  ( rec ( F ,  A
)  |`  B ) )  ->  dom  g  =  B )
18 fveq1 5388 . . . . . . 7  |-  ( g  =  ( rec ( F ,  A )  |`  B )  ->  (
g `  x )  =  ( ( rec ( F ,  A
)  |`  B ) `  x ) )
1918fveq2d 5393 . . . . . 6  |-  ( g  =  ( rec ( F ,  A )  |`  B )  ->  ( F `  ( g `  x ) )  =  ( F `  (
( rec ( F ,  A )  |`  B ) `  x
) ) )
2019adantl 275 . . . . 5  |-  ( ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  /\  g  =  ( rec ( F ,  A
)  |`  B ) )  ->  ( F `  ( g `  x
) )  =  ( F `  ( ( rec ( F ,  A )  |`  B ) `
 x ) ) )
2117, 20iuneq12d 3807 . . . 4  |-  ( ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  /\  g  =  ( rec ( F ,  A
)  |`  B ) )  ->  U_ x  e.  dom  g ( F `  ( g `  x
) )  =  U_ x  e.  B  ( F `  ( ( rec ( F ,  A
)  |`  B ) `  x ) ) )
2221uneq2d 3200 . . 3  |-  ( ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  /\  g  =  ( rec ( F ,  A
)  |`  B ) )  ->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) )  =  ( A  u.  U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `  x
) ) ) )
23 rdgfun 6238 . . . . 5  |-  Fun  rec ( F ,  A )
24 resfunexg 5609 . . . . 5  |-  ( ( Fun  rec ( F ,  A )  /\  B  e.  On )  ->  ( rec ( F ,  A )  |`  B )  e.  _V )
2523, 24mpan 420 . . . 4  |-  ( B  e.  On  ->  ( rec ( F ,  A
)  |`  B )  e. 
_V )
26253ad2ant3 989 . . 3  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  ( rec ( F ,  A )  |`  B )  e.  _V )
27 simpr 109 . . . . . 6  |-  ( ( F  Fn  _V  /\  B  e.  On )  ->  B  e.  On )
28 vex 2663 . . . . . . . . . 10  |-  x  e. 
_V
29 fvexg 5408 . . . . . . . . . 10  |-  ( ( ( rec ( F ,  A )  |`  B )  e.  _V  /\  x  e.  _V )  ->  ( ( rec ( F ,  A )  |`  B ) `  x
)  e.  _V )
3025, 28, 29sylancl 409 . . . . . . . . 9  |-  ( B  e.  On  ->  (
( rec ( F ,  A )  |`  B ) `  x
)  e.  _V )
3130ralrimivw 2483 . . . . . . . 8  |-  ( B  e.  On  ->  A. x  e.  B  ( ( rec ( F ,  A
)  |`  B ) `  x )  e.  _V )
3231adantl 275 . . . . . . 7  |-  ( ( F  Fn  _V  /\  B  e.  On )  ->  A. x  e.  B  ( ( rec ( F ,  A )  |`  B ) `  x
)  e.  _V )
33 funfvex 5406 . . . . . . . . . . 11  |-  ( ( Fun  F  /\  (
( rec ( F ,  A )  |`  B ) `  x
)  e.  dom  F
)  ->  ( F `  ( ( rec ( F ,  A )  |`  B ) `  x
) )  e.  _V )
3433funfni 5193 . . . . . . . . . 10  |-  ( ( F  Fn  _V  /\  ( ( rec ( F ,  A )  |`  B ) `  x
)  e.  _V )  ->  ( F `  (
( rec ( F ,  A )  |`  B ) `  x
) )  e.  _V )
3534ex 114 . . . . . . . . 9  |-  ( F  Fn  _V  ->  (
( ( rec ( F ,  A )  |`  B ) `  x
)  e.  _V  ->  ( F `  ( ( rec ( F ,  A )  |`  B ) `
 x ) )  e.  _V ) )
3635ralimdv 2477 . . . . . . . 8  |-  ( F  Fn  _V  ->  ( A. x  e.  B  ( ( rec ( F ,  A )  |`  B ) `  x
)  e.  _V  ->  A. x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `
 x ) )  e.  _V ) )
3736adantr 274 . . . . . . 7  |-  ( ( F  Fn  _V  /\  B  e.  On )  ->  ( A. x  e.  B  ( ( rec ( F ,  A
)  |`  B ) `  x )  e.  _V  ->  A. x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `
 x ) )  e.  _V ) )
3832, 37mpd 13 . . . . . 6  |-  ( ( F  Fn  _V  /\  B  e.  On )  ->  A. x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `
 x ) )  e.  _V )
39 iunexg 5985 . . . . . 6  |-  ( ( B  e.  On  /\  A. x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `
 x ) )  e.  _V )  ->  U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `
 x ) )  e.  _V )
4027, 38, 39syl2anc 408 . . . . 5  |-  ( ( F  Fn  _V  /\  B  e.  On )  ->  U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `
 x ) )  e.  _V )
41403adant2 985 . . . 4  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `
 x ) )  e.  _V )
42 unexg 4334 . . . . . 6  |-  ( ( A  e.  V  /\  U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `
 x ) )  e.  _V )  -> 
( A  u.  U_ x  e.  B  ( F `  ( ( rec ( F ,  A
)  |`  B ) `  x ) ) )  e.  _V )
4342ex 114 . . . . 5  |-  ( A  e.  V  ->  ( U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `
 x ) )  e.  _V  ->  ( A  u.  U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `  x
) ) )  e. 
_V ) )
44433ad2ant2 988 . . . 4  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  ( U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `  x
) )  e.  _V  ->  ( A  u.  U_ x  e.  B  ( F `  ( ( rec ( F ,  A
)  |`  B ) `  x ) ) )  e.  _V ) )
4541, 44mpd 13 . . 3  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  ( A  u.  U_ x  e.  B  ( F `  ( ( rec ( F ,  A
)  |`  B ) `  x ) ) )  e.  _V )
466, 22, 26, 45fvmptd 5470 . 2  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) `  ( rec ( F ,  A )  |`  B ) )  =  ( A  u.  U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `  x
) ) ) )
475, 46eqtrd 2150 1  |-  ( ( F  Fn  _V  /\  A  e.  V  /\  B  e.  On )  ->  ( rec ( F ,  A ) `  B )  =  ( A  u.  U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `  x
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316    e. wcel 1465   A.wral 2393   _Vcvv 2660    u. cun 3039    C_ wss 3041   U_ciun 3783    |-> cmpt 3959   Oncon0 4255   dom cdm 4509    |` cres 4511   Fun wfun 5087    Fn wfn 5088   ` cfv 5093   reccrdg 6234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-recs 6170  df-irdg 6235
This theorem is referenced by:  rdgival  6247
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