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Theorem rdg0 6347
Description: The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypothesis
Ref Expression
rdg.1  |-  A  e. 
_V
Assertion
Ref Expression
rdg0  |-  ( rec ( F ,  A
) `  (/) )  =  A

Proof of Theorem rdg0
Dummy variables  x  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4104 . . . . 5  |-  (/)  e.  _V
2 dmeq 4799 . . . . . . . 8  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
3 fveq1 5480 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( g `
 x )  =  ( (/) `  x ) )
43fveq2d 5485 . . . . . . . 8  |-  ( g  =  (/)  ->  ( F `
 ( g `  x ) )  =  ( F `  ( (/) `  x ) ) )
52, 4iuneq12d 3885 . . . . . . 7  |-  ( g  =  (/)  ->  U_ x  e.  dom  g ( F `
 ( g `  x ) )  = 
U_ x  e.  dom  (/) ( F `  ( (/) `  x ) ) )
65uneq2d 3272 . . . . . 6  |-  ( g  =  (/)  ->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )  =  ( A  u.  U_ x  e.  dom  (/) ( F `
 ( (/) `  x
) ) ) )
7 eqid 2164 . . . . . 6  |-  ( 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 ) ) ) )
8 rdg.1 . . . . . . 7  |-  A  e. 
_V
9 dm0 4813 . . . . . . . . . 10  |-  dom  (/)  =  (/)
10 iuneq1 3874 . . . . . . . . . 10  |-  ( dom  (/)  =  (/)  ->  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  =  U_ x  e.  (/)  ( F `  ( (/) `  x ) ) )
119, 10ax-mp 5 . . . . . . . . 9  |-  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  =  U_ x  e.  (/)  ( F `  ( (/) `  x ) )
12 0iun 3918 . . . . . . . . 9  |-  U_ x  e.  (/)  ( F `  ( (/) `  x ) )  =  (/)
1311, 12eqtri 2185 . . . . . . . 8  |-  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  =  (/)
1413, 1eqeltri 2237 . . . . . . 7  |-  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  e.  _V
158, 14unex 4414 . . . . . 6  |-  ( A  u.  U_ x  e. 
dom  (/) ( F `  ( (/) `  x ) ) )  e.  _V
166, 7, 15fvmpt 5558 . . . . 5  |-  ( (/)  e.  _V  ->  ( (
g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  =  ( A  u.  U_ x  e.  dom  (/) ( F `
 ( (/) `  x
) ) ) )
171, 16ax-mp 5 . . . 4  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  =  ( A  u.  U_ x  e.  dom  (/) ( F `
 ( (/) `  x
) ) )
1817, 15eqeltri 2237 . . 3  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  e. 
_V
19 df-irdg 6330 . . . 4  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
2019tfr0 6283 . . 3  |-  ( ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  e. 
_V  ->  ( rec ( F ,  A ) `  (/) )  =  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) ) )
2118, 20ax-mp 5 . 2  |-  ( rec ( F ,  A
) `  (/) )  =  ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) `  (/) )
2213uneq2i 3269 . . . 4  |-  ( A  u.  U_ x  e. 
dom  (/) ( F `  ( (/) `  x ) ) )  =  ( A  u.  (/) )
2317, 22eqtri 2185 . . 3  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  =  ( A  u.  (/) )
24 un0 3438 . . 3  |-  ( A  u.  (/) )  =  A
2523, 24eqtri 2185 . 2  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  =  A
2621, 25eqtri 2185 1  |-  ( rec ( F ,  A
) `  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1342    e. wcel 2135   _Vcvv 2722    u. cun 3110   (/)c0 3405   U_ciun 3861    |-> cmpt 4038   dom cdm 4599   ` cfv 5183   reccrdg 6329
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2724  df-sbc 2948  df-csb 3042  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-iun 3863  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-id 4266  df-iord 4339  df-on 4341  df-suc 4344  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-res 4611  df-iota 5148  df-fun 5185  df-fn 5186  df-fv 5191  df-recs 6265  df-irdg 6330
This theorem is referenced by:  rdg0g  6348  om0  6418
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