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Theorem rdg0 6134
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 3958 . . . . 5  |-  (/)  e.  _V
2 dmeq 4624 . . . . . . . 8  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
3 fveq1 5288 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( g `
 x )  =  ( (/) `  x ) )
43fveq2d 5293 . . . . . . . 8  |-  ( g  =  (/)  ->  ( F `
 ( g `  x ) )  =  ( F `  ( (/) `  x ) ) )
52, 4iuneq12d 3749 . . . . . . 7  |-  ( g  =  (/)  ->  U_ x  e.  dom  g ( F `
 ( g `  x ) )  = 
U_ x  e.  dom  (/) ( F `  ( (/) `  x ) ) )
65uneq2d 3152 . . . . . 6  |-  ( g  =  (/)  ->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )  =  ( A  u.  U_ x  e.  dom  (/) ( F `
 ( (/) `  x
) ) ) )
7 eqid 2088 . . . . . 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 4638 . . . . . . . . . 10  |-  dom  (/)  =  (/)
10 iuneq1 3738 . . . . . . . . . 10  |-  ( dom  (/)  =  (/)  ->  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  =  U_ x  e.  (/)  ( F `  ( (/) `  x ) ) )
119, 10ax-mp 7 . . . . . . . . 9  |-  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  =  U_ x  e.  (/)  ( F `  ( (/) `  x ) )
12 0iun 3782 . . . . . . . . 9  |-  U_ x  e.  (/)  ( F `  ( (/) `  x ) )  =  (/)
1311, 12eqtri 2108 . . . . . . . 8  |-  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  =  (/)
1413, 1eqeltri 2160 . . . . . . 7  |-  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  e.  _V
158, 14unex 4257 . . . . . 6  |-  ( A  u.  U_ x  e. 
dom  (/) ( F `  ( (/) `  x ) ) )  e.  _V
166, 7, 15fvmpt 5365 . . . . 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 7 . . . 4  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  =  ( A  u.  U_ x  e.  dom  (/) ( F `
 ( (/) `  x
) ) )
1817, 15eqeltri 2160 . . 3  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  e. 
_V
19 df-irdg 6117 . . . 4  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
2019tfr0 6070 . . 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 7 . 2  |-  ( rec ( F ,  A
) `  (/) )  =  ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) `  (/) )
2213uneq2i 3149 . . . 4  |-  ( A  u.  U_ x  e. 
dom  (/) ( F `  ( (/) `  x ) ) )  =  ( A  u.  (/) )
2317, 22eqtri 2108 . . 3  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  =  ( A  u.  (/) )
24 un0 3314 . . 3  |-  ( A  u.  (/) )  =  A
2523, 24eqtri 2108 . 2  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  =  A
2621, 25eqtri 2108 1  |-  ( rec ( F ,  A
) `  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   _Vcvv 2619    u. cun 2995   (/)c0 3284   U_ciun 3725    |-> cmpt 3891   dom cdm 4428   ` cfv 5002   reccrdg 6116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-res 4440  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010  df-recs 6052  df-irdg 6117
This theorem is referenced by:  rdg0g  6135  om0  6201
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