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Theorem rdgss 6386
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 3151 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
3 ssid 3177 . . . . . . 7  |-  ( F `
 ( rec ( F ,  I ) `  x ) )  C_  ( F `  ( rec ( F ,  I
) `  x )
)
4 fveq2 5517 . . . . . . . . . 10  |-  ( y  =  x  ->  ( rec ( F ,  I
) `  y )  =  ( rec ( F ,  I ) `  x ) )
54fveq2d 5521 . . . . . . . . 9  |-  ( y  =  x  ->  ( F `  ( rec ( F ,  I ) `
 y ) )  =  ( F `  ( rec ( F ,  I ) `  x
) ) )
65sseq2d 3187 . . . . . . . 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 2843 . . . . . . 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 425 . . . . . 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 2549 . . . 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 3933 . . 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 3308 . . 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 6385 . . 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 1238 . 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 6385 . . 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 1238 . 2  |-  ( ph  ->  ( rec ( F ,  I ) `  B )  =  ( I  u.  U_ y  e.  B  ( F `  ( rec ( F ,  I ) `  y ) ) ) )
2314, 19, 223sstr4d 3202 1  |-  ( ph  ->  ( rec ( F ,  I ) `  A )  C_  ( rec ( F ,  I
) `  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2739    u. cun 3129    C_ wss 3131   U_ciun 3888   Oncon0 4365    Fn wfn 5213   ` cfv 5218   reccrdg 6372
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 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-recs 6308  df-irdg 6373
This theorem is referenced by:  oawordi  6472
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