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Theorem rdg0 6036
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 3913 . . . . 5  |-  (/)  e.  _V
2 dmeq 4563 . . . . . . . 8  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
3 fveq1 5208 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( g `
 x )  =  ( (/) `  x ) )
43fveq2d 5213 . . . . . . . 8  |-  ( g  =  (/)  ->  ( F `
 ( g `  x ) )  =  ( F `  ( (/) `  x ) ) )
52, 4iuneq12d 3710 . . . . . . 7  |-  ( g  =  (/)  ->  U_ x  e.  dom  g ( F `
 ( g `  x ) )  = 
U_ x  e.  dom  (/) ( F `  ( (/) `  x ) ) )
65uneq2d 3127 . . . . . 6  |-  ( g  =  (/)  ->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )  =  ( A  u.  U_ x  e.  dom  (/) ( F `
 ( (/) `  x
) ) ) )
7 eqid 2082 . . . . . 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 4577 . . . . . . . . . 10  |-  dom  (/)  =  (/)
10 iuneq1 3699 . . . . . . . . . 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 3743 . . . . . . . . 9  |-  U_ x  e.  (/)  ( F `  ( (/) `  x ) )  =  (/)
1311, 12eqtri 2102 . . . . . . . 8  |-  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  =  (/)
1413, 1eqeltri 2152 . . . . . . 7  |-  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  e.  _V
158, 14unex 4202 . . . . . 6  |-  ( A  u.  U_ x  e. 
dom  (/) ( F `  ( (/) `  x ) ) )  e.  _V
166, 7, 15fvmpt 5281 . . . . 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 2152 . . 3  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  e. 
_V
19 df-irdg 6019 . . . 4  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
2019tfr0 5972 . . 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 3124 . . . 4  |-  ( A  u.  U_ x  e. 
dom  (/) ( F `  ( (/) `  x ) ) )  =  ( A  u.  (/) )
2317, 22eqtri 2102 . . 3  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  =  ( A  u.  (/) )
24 un0 3285 . . 3  |-  ( A  u.  (/) )  =  A
2523, 24eqtri 2102 . 2  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  =  A
2621, 25eqtri 2102 1  |-  ( rec ( F ,  A
) `  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   _Vcvv 2602    u. cun 2972   (/)c0 3258   U_ciun 3686    |-> cmpt 3847   dom cdm 4371   ` cfv 4932   reccrdg 6018
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-res 4383  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940  df-recs 5954  df-irdg 6019
This theorem is referenced by:  rdg0g  6037  om0  6102
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