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Theorem dffrege76 36505
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 36289 . . 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 1360 . 2  |-  ( B ( t+ `  R ) E  <->  E  e.  |^|
{ f  |  ( R " ( { B }  u.  f
) )  C_  f } )
62elexi 3090 . . 3  |-  E  e. 
_V
76elintab 4266 . 2  |-  ( E  e.  |^| { f  |  ( R " ( { B }  u.  f
) )  C_  f } 
<-> 
A. f ( ( R " ( { B }  u.  f
) )  C_  f  ->  E  e.  f ) )
8 imaundi 5267 . . . . . . . . 9  |-  ( R
" ( { B }  u.  f )
)  =  ( ( R " { B } )  u.  ( R " f ) )
98equncomi 3612 . . . . . . . 8  |-  ( R
" ( { B }  u.  f )
)  =  ( ( R " f )  u.  ( R " { B } ) )
109sseq1i 3488 . . . . . . 7  |-  ( ( R " ( { B }  u.  f
) )  C_  f  <->  ( ( R " f
)  u.  ( R
" { B }
) )  C_  f
)
11 unss 3640 . . . . . . 7  |-  ( ( ( R " f
)  C_  f  /\  ( R " { B } )  C_  f
)  <->  ( ( R
" f )  u.  ( R " { B } ) )  C_  f )
1210, 11bitr4i 255 . . . . . 6  |-  ( ( R " ( { B }  u.  f
) )  C_  f  <->  ( ( R " f
)  C_  f  /\  ( R " { B } )  C_  f
) )
13 df-he 36338 . . . . . . . 8  |-  ( R hereditary  f 
<->  ( R " f
)  C_  f )
1413bicomi 205 . . . . . . 7  |-  ( ( R " f ) 
C_  f  <->  R hereditary  f )
15 dfss2 3453 . . . . . . . 8  |-  ( ( R " { B } )  C_  f  <->  A. a ( a  e.  ( R " { B } )  ->  a  e.  f ) )
161elexi 3090 . . . . . . . . . . . 12  |-  B  e. 
_V
17 vex 3083 . . . . . . . . . . . 12  |-  a  e. 
_V
1816, 17elimasn 5212 . . . . . . . . . . 11  |-  ( a  e.  ( R " { B } )  <->  <. B , 
a >.  e.  R )
19 df-br 4424 . . . . . . . . . . 11  |-  ( B R a  <->  <. B , 
a >.  e.  R )
2018, 19bitr4i 255 . . . . . . . . . 10  |-  ( a  e.  ( R " { B } )  <->  B R
a )
2120imbi1i 326 . . . . . . . . 9  |-  ( ( a  e.  ( R
" { B }
)  ->  a  e.  f )  <->  ( B R a  ->  a  e.  f ) )
2221albii 1685 . . . . . . . 8  |-  ( A. a ( a  e.  ( R " { B } )  ->  a  e.  f )  <->  A. a
( B R a  ->  a  e.  f ) )
2315, 22bitri 252 . . . . . . 7  |-  ( ( R " { B } )  C_  f  <->  A. a ( B R a  ->  a  e.  f ) )
2414, 23anbi12i 701 . . . . . 6  |-  ( ( ( R " f
)  C_  f  /\  ( R " { B } )  C_  f
)  <->  ( R hereditary  f  /\  A. a ( B R a  ->  a  e.  f ) ) )
2512, 24bitri 252 . . . . 5  |-  ( ( R " ( { B }  u.  f
) )  C_  f  <->  ( R hereditary  f  /\  A. a
( B R a  ->  a  e.  f ) ) )
2625imbi1i 326 . . . 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 447 . . . 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 252 . . 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 1685 . 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 275 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 187    /\ wa 370   A.wal 1435    e. wcel 1872   {cab 2407    u. cun 3434    C_ wss 3436   {csn 3998   <.cop 4004   |^|cint 4255   class class class wbr 4423   "cima 4856   ` cfv 5601   t+ctcl 13049   hereditary whe 36337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-seq 12220  df-trcl 13051  df-relexp 13084  df-he 36338
This theorem is referenced by:  frege77  36506  frege89  36518
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