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Theorem tfrex 6363
Description: The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypotheses
Ref Expression
tfrex.1  |-  F  = recs ( G )
tfrex.2  |-  ( ph  ->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
Assertion
Ref Expression
tfrex  |-  ( (
ph  /\  A  e.  V )  ->  ( F `  A )  e.  _V )
Distinct variable group:    x, G
Allowed substitution hints:    ph( x)    A( x)    F( x)    V( x)

Proof of Theorem tfrex
Dummy variables  f  g  u  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrex.1 . . 3  |-  F  = recs ( G )
21fveq1i 5512 . 2  |-  ( F `
 A )  =  (recs ( G ) `
 A )
3 eqid 2177 . . . 4  |-  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( G `  ( g  |`  u ) ) ) }  =  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( G `  ( g  |`  u ) ) ) }
43tfrlem3 6306 . . 3  |-  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. u  e.  z  ( g `  u )  =  ( G `  ( g  |`  u ) ) ) }  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
5 tfrex.2 . . 3  |-  ( ph  ->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
64, 5tfrexlem 6329 . 2  |-  ( (
ph  /\  A  e.  V )  ->  (recs ( G ) `  A
)  e.  _V )
72, 6eqeltrid 2264 1  |-  ( (
ph  /\  A  e.  V )  ->  ( F `  A )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1351    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2737   Oncon0 4360    |` cres 4625   Fun wfun 5206    Fn wfn 5207   ` cfv 5212  recscrecs 6299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-recs 6300
This theorem is referenced by:  rdgexggg  6372
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