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Theorem dffrege76 36606
Description: If from the two propositions that every result of an application of the procedure  R to  B has property  f and that property  f is hereditary in the  R-sequence, it can be inferred, whatever  f may be, that  E has property  f, then we say  E follows  B in the  R-sequence. Definition 76 of [Frege1879] p. 60.

Each of  B,  E and  R must be sets. (Contributed by RP, 2-Jul-2020.)

Hypotheses
Ref Expression
frege76.b  |-  B  e.  U
frege76.e  |-  E  e.  V
frege76.r  |-  R  e.  W
Assertion
Ref Expression
dffrege76  |-  ( A. f ( R hereditary  f  -> 
( A. a ( B R a  -> 
a  e.  f )  ->  E  e.  f ) )  <->  B (
t+ `  R
) E )
Distinct variable groups:    f, a, B    f, E    R, a,
f    U, f    f, V   
f, W
Allowed substitution hints:    U( a)    E( a)    V( a)    W( a)

Proof of Theorem dffrege76
StepHypRef Expression
1 frege76.b . . 3  |-  B  e.  U
2 frege76.e . . 3  |-  E  e.  V
3 frege76.r . . 3  |-  R  e.  W
4 brtrclfv2 36390 . . 3  |-  ( ( B  e.  U  /\  E  e.  V  /\  R  e.  W )  ->  ( B ( t+ `  R ) E  <->  E  e.  |^| { f  |  ( R "
( { B }  u.  f ) )  C_  f } ) )
51, 2, 3, 4mp3an 1390 . 2  |-  ( B ( t+ `  R ) E  <->  E  e.  |^|
{ f  |  ( R " ( { B }  u.  f
) )  C_  f } )
62elexi 3041 . . 3  |-  E  e. 
_V
76elintab 4237 . 2  |-  ( E  e.  |^| { f  |  ( R " ( { B }  u.  f
) )  C_  f } 
<-> 
A. f ( ( R " ( { B }  u.  f
) )  C_  f  ->  E  e.  f ) )
8 imaundi 5254 . . . . . . . . 9  |-  ( R
" ( { B }  u.  f )
)  =  ( ( R " { B } )  u.  ( R " f ) )
98equncomi 3571 . . . . . . . 8  |-  ( R
" ( { B }  u.  f )
)  =  ( ( R " f )  u.  ( R " { B } ) )
109sseq1i 3442 . . . . . . 7  |-  ( ( R " ( { B }  u.  f
) )  C_  f  <->  ( ( R " f
)  u.  ( R
" { B }
) )  C_  f
)
11 unss 3599 . . . . . . 7  |-  ( ( ( R " f
)  C_  f  /\  ( R " { B } )  C_  f
)  <->  ( ( R
" f )  u.  ( R " { B } ) )  C_  f )
1210, 11bitr4i 260 . . . . . 6  |-  ( ( R " ( { B }  u.  f
) )  C_  f  <->  ( ( R " f
)  C_  f  /\  ( R " { B } )  C_  f
) )
13 df-he 36439 . . . . . . . 8  |-  ( R hereditary  f 
<->  ( R " f
)  C_  f )
1413bicomi 207 . . . . . . 7  |-  ( ( R " f ) 
C_  f  <->  R hereditary  f )
15 dfss2 3407 . . . . . . . 8  |-  ( ( R " { B } )  C_  f  <->  A. a ( a  e.  ( R " { B } )  ->  a  e.  f ) )
161elexi 3041 . . . . . . . . . . . 12  |-  B  e. 
_V
17 vex 3034 . . . . . . . . . . . 12  |-  a  e. 
_V
1816, 17elimasn 5199 . . . . . . . . . . 11  |-  ( a  e.  ( R " { B } )  <->  <. B , 
a >.  e.  R )
19 df-br 4396 . . . . . . . . . . 11  |-  ( B R a  <->  <. B , 
a >.  e.  R )
2018, 19bitr4i 260 . . . . . . . . . 10  |-  ( a  e.  ( R " { B } )  <->  B R
a )
2120imbi1i 332 . . . . . . . . 9  |-  ( ( a  e.  ( R
" { B }
)  ->  a  e.  f )  <->  ( B R a  ->  a  e.  f ) )
2221albii 1699 . . . . . . . 8  |-  ( A. a ( a  e.  ( R " { B } )  ->  a  e.  f )  <->  A. a
( B R a  ->  a  e.  f ) )
2315, 22bitri 257 . . . . . . 7  |-  ( ( R " { B } )  C_  f  <->  A. a ( B R a  ->  a  e.  f ) )
2414, 23anbi12i 711 . . . . . 6  |-  ( ( ( R " f
)  C_  f  /\  ( R " { B } )  C_  f
)  <->  ( R hereditary  f  /\  A. a ( B R a  ->  a  e.  f ) ) )
2512, 24bitri 257 . . . . 5  |-  ( ( R " ( { B }  u.  f
) )  C_  f  <->  ( R hereditary  f  /\  A. a
( B R a  ->  a  e.  f ) ) )
2625imbi1i 332 . . . 4  |-  ( ( ( R " ( { B }  u.  f
) )  C_  f  ->  E  e.  f )  <-> 
( ( R hereditary  f  /\  A. a ( B R a  ->  a  e.  f ) )  ->  E  e.  f )
)
27 impexp 453 . . . 4  |-  ( ( ( R hereditary  f  /\  A. a ( B R a  ->  a  e.  f ) )  ->  E  e.  f )  <->  ( R hereditary  f  ->  ( A. a ( B R a  ->  a  e.  f )  ->  E  e.  f ) ) )
2826, 27bitri 257 . . 3  |-  ( ( ( R " ( { B }  u.  f
) )  C_  f  ->  E  e.  f )  <-> 
( R hereditary  f  ->  ( A. a ( B R a  ->  a  e.  f )  ->  E  e.  f ) ) )
2928albii 1699 . 2  |-  ( A. f ( ( R
" ( { B }  u.  f )
)  C_  f  ->  E  e.  f )  <->  A. f
( R hereditary  f  ->  ( A. a ( B R a  ->  a  e.  f )  ->  E  e.  f ) ) )
305, 7, 293bitrri 280 1  |-  ( A. f ( R hereditary  f  -> 
( A. a ( B R a  -> 
a  e.  f )  ->  E  e.  f ) )  <->  B (
t+ `  R
) E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    e. wcel 1904   {cab 2457    u. cun 3388    C_ wss 3390   {csn 3959   <.cop 3965   |^|cint 4226   class class class wbr 4395   "cima 4842   ` cfv 5589   t+ctcl 13124   hereditary whe 36438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-seq 12252  df-trcl 13126  df-relexp 13161  df-he 36439
This theorem is referenced by:  frege77  36607  frege89  36619
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