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Theorem rdg0 5974
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 3884 . . . . 5  |-  (/)  e.  _V
2 dmeq 4535 . . . . . . . 8  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
3 fveq1 5177 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( g `
 x )  =  ( (/) `  x ) )
43fveq2d 5182 . . . . . . . 8  |-  ( g  =  (/)  ->  ( F `
 ( g `  x ) )  =  ( F `  ( (/) `  x ) ) )
52, 4iuneq12d 3681 . . . . . . 7  |-  ( g  =  (/)  ->  U_ x  e.  dom  g ( F `
 ( g `  x ) )  = 
U_ x  e.  dom  (/) ( F `  ( (/) `  x ) ) )
65uneq2d 3097 . . . . . 6  |-  ( g  =  (/)  ->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )  =  ( A  u.  U_ x  e.  dom  (/) ( F `
 ( (/) `  x
) ) ) )
7 eqid 2040 . . . . . 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 4549 . . . . . . . . . 10  |-  dom  (/)  =  (/)
10 iuneq1 3670 . . . . . . . . . 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 3714 . . . . . . . . 9  |-  U_ x  e.  (/)  ( F `  ( (/) `  x ) )  =  (/)
1311, 12eqtri 2060 . . . . . . . 8  |-  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  =  (/)
1413, 1eqeltri 2110 . . . . . . 7  |-  U_ x  e.  dom  (/) ( F `  ( (/) `  x ) )  e.  _V
158, 14unex 4176 . . . . . 6  |-  ( A  u.  U_ x  e. 
dom  (/) ( F `  ( (/) `  x ) ) )  e.  _V
166, 7, 15fvmpt 5249 . . . . 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 2110 . . 3  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  e. 
_V
19 df-irdg 5957 . . . 4  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
2019tfr0 5937 . . 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 3094 . . . 4  |-  ( A  u.  U_ x  e. 
dom  (/) ( F `  ( (/) `  x ) ) )  =  ( A  u.  (/) )
2317, 22eqtri 2060 . . 3  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  =  ( A  u.  (/) )
24 un0 3251 . . 3  |-  ( A  u.  (/) )  =  A
2523, 24eqtri 2060 . 2  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  (/) )  =  A
2621, 25eqtri 2060 1  |-  ( rec ( F ,  A
) `  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1243    e. wcel 1393   _Vcvv 2557    u. cun 2915   (/)c0 3224   U_ciun 3657    |-> cmpt 3818   dom cdm 4345   ` cfv 4902   reccrdg 5956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-recs 5920  df-irdg 5957
This theorem is referenced by:  rdg0g  5975  om0  6038
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