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Theorem rdgss 6248
Description: Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)
Hypotheses
Ref Expression
rdgss.1  |-  ( ph  ->  F  Fn  _V )
rdgss.2  |-  ( ph  ->  I  e.  V )
rdgss.3  |-  ( ph  ->  A  e.  On )
rdgss.4  |-  ( ph  ->  B  e.  On )
rdgss.5  |-  ( ph  ->  A  C_  B )
Assertion
Ref Expression
rdgss  |-  ( ph  ->  ( rec ( F ,  I ) `  A )  C_  ( rec ( F ,  I
) `  B )
)

Proof of Theorem rdgss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rdgss.5 . . . 4  |-  ( ph  ->  A  C_  B )
2 ssel 3061 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
3 ssid 3087 . . . . . . 7  |-  ( F `
 ( rec ( F ,  I ) `  x ) )  C_  ( F `  ( rec ( F ,  I
) `  x )
)
4 fveq2 5389 . . . . . . . . . 10  |-  ( y  =  x  ->  ( rec ( F ,  I
) `  y )  =  ( rec ( F ,  I ) `  x ) )
54fveq2d 5393 . . . . . . . . 9  |-  ( y  =  x  ->  ( F `  ( rec ( F ,  I ) `
 y ) )  =  ( F `  ( rec ( F ,  I ) `  x
) ) )
65sseq2d 3097 . . . . . . . 8  |-  ( y  =  x  ->  (
( F `  ( rec ( F ,  I
) `  x )
)  C_  ( F `  ( rec ( F ,  I ) `  y ) )  <->  ( F `  ( rec ( F ,  I ) `  x ) )  C_  ( F `  ( rec ( F ,  I
) `  x )
) ) )
76rspcev 2763 . . . . . . 7  |-  ( ( x  e.  B  /\  ( F `  ( rec ( F ,  I
) `  x )
)  C_  ( F `  ( rec ( F ,  I ) `  x ) ) )  ->  E. y  e.  B  ( F `  ( rec ( F ,  I
) `  x )
)  C_  ( F `  ( rec ( F ,  I ) `  y ) ) )
83, 7mpan2 421 . . . . . 6  |-  ( x  e.  B  ->  E. y  e.  B  ( F `  ( rec ( F ,  I ) `  x ) )  C_  ( F `  ( rec ( F ,  I
) `  y )
) )
92, 8syl6 33 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  ->  E. y  e.  B  ( F `  ( rec ( F ,  I
) `  x )
)  C_  ( F `  ( rec ( F ,  I ) `  y ) ) ) )
109ralrimiv 2481 . . . 4  |-  ( A 
C_  B  ->  A. x  e.  A  E. y  e.  B  ( F `  ( rec ( F ,  I ) `  x ) )  C_  ( F `  ( rec ( F ,  I
) `  y )
) )
111, 10syl 14 . . 3  |-  ( ph  ->  A. x  e.  A  E. y  e.  B  ( F `  ( rec ( F ,  I
) `  x )
)  C_  ( F `  ( rec ( F ,  I ) `  y ) ) )
12 iunss2 3828 . . 3  |-  ( A. x  e.  A  E. y  e.  B  ( F `  ( rec ( F ,  I ) `
 x ) ) 
C_  ( F `  ( rec ( F ,  I ) `  y
) )  ->  U_ x  e.  A  ( F `  ( rec ( F ,  I ) `  x ) )  C_  U_ y  e.  B  ( F `  ( rec ( F ,  I
) `  y )
) )
13 unss2 3217 . . 3  |-  ( U_ x  e.  A  ( F `  ( rec ( F ,  I ) `
 x ) ) 
C_  U_ y  e.  B  ( F `  ( rec ( F ,  I
) `  y )
)  ->  ( I  u.  U_ x  e.  A  ( F `  ( rec ( F ,  I
) `  x )
) )  C_  (
I  u.  U_ y  e.  B  ( F `  ( rec ( F ,  I ) `  y ) ) ) )
1411, 12, 133syl 17 . 2  |-  ( ph  ->  ( I  u.  U_ x  e.  A  ( F `  ( rec ( F ,  I ) `
 x ) ) )  C_  ( I  u.  U_ y  e.  B  ( F `  ( rec ( F ,  I
) `  y )
) ) )
15 rdgss.1 . . 3  |-  ( ph  ->  F  Fn  _V )
16 rdgss.2 . . 3  |-  ( ph  ->  I  e.  V )
17 rdgss.3 . . 3  |-  ( ph  ->  A  e.  On )
18 rdgival 6247 . . 3  |-  ( ( F  Fn  _V  /\  I  e.  V  /\  A  e.  On )  ->  ( rec ( F ,  I ) `  A )  =  ( I  u.  U_ x  e.  A  ( F `  ( rec ( F ,  I ) `  x ) ) ) )
1915, 16, 17, 18syl3anc 1201 . 2  |-  ( ph  ->  ( rec ( F ,  I ) `  A )  =  ( I  u.  U_ x  e.  A  ( F `  ( rec ( F ,  I ) `  x ) ) ) )
20 rdgss.4 . . 3  |-  ( ph  ->  B  e.  On )
21 rdgival 6247 . . 3  |-  ( ( F  Fn  _V  /\  I  e.  V  /\  B  e.  On )  ->  ( rec ( F ,  I ) `  B )  =  ( I  u.  U_ y  e.  B  ( F `  ( rec ( F ,  I ) `  y ) ) ) )
2215, 16, 20, 21syl3anc 1201 . 2  |-  ( ph  ->  ( rec ( F ,  I ) `  B )  =  ( I  u.  U_ y  e.  B  ( F `  ( rec ( F ,  I ) `  y ) ) ) )
2314, 19, 223sstr4d 3112 1  |-  ( ph  ->  ( rec ( F ,  I ) `  A )  C_  ( rec ( F ,  I
) `  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   _Vcvv 2660    u. cun 3039    C_ wss 3041   U_ciun 3783   Oncon0 4255    Fn wfn 5088   ` cfv 5093   reccrdg 6234
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-recs 6170  df-irdg 6235
This theorem is referenced by:  oawordi  6333
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