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Theorem frege124d 36219
Description: If  F is a function,  A is the successor of  X, and  B follows  X in the transitive closure of  F, then  A and  B are the same or  B follows  A in the transitive closure of  F. Similar to Proposition 124 of [Frege1879] p. 80. Compare with frege124 36447. (Contributed by RP, 16-Jul-2020.)
Hypotheses
Ref Expression
frege124d.f  |-  ( ph  ->  F  e.  _V )
frege124d.x  |-  ( ph  ->  X  e.  dom  F
)
frege124d.a  |-  ( ph  ->  A  =  ( F `
 X ) )
frege124d.xb  |-  ( ph  ->  X ( t+ `  F ) B )
frege124d.fun  |-  ( ph  ->  Fun  F )
Assertion
Ref Expression
frege124d  |-  ( ph  ->  ( A ( t+ `  F ) B  \/  A  =  B ) )

Proof of Theorem frege124d
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 frege124d.a . . 3  |-  ( ph  ->  A  =  ( F `
 X ) )
2 frege124d.fun . . . . 5  |-  ( ph  ->  Fun  F )
3 frege124d.xb . . . . . . 7  |-  ( ph  ->  X ( t+ `  F ) B )
41eqcomd 2431 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  X
)  =  A )
5 frege124d.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  dom  F
)
6 funbrfvb 5921 . . . . . . . . . . . 12  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( ( F `  X )  =  A  <-> 
X F A ) )
72, 5, 6syl2anc 666 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F `  X )  =  A  <-> 
X F A ) )
84, 7mpbid 214 . . . . . . . . . 10  |-  ( ph  ->  X F A )
9 funeu 5623 . . . . . . . . . 10  |-  ( ( Fun  F  /\  X F A )  ->  E! a  X F a )
102, 8, 9syl2anc 666 . . . . . . . . 9  |-  ( ph  ->  E! a  X F a )
11 fvex 5889 . . . . . . . . . . . . 13  |-  ( F `
 X )  e. 
_V
121, 11syl6eqel 2519 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  _V )
13 sbcan 3343 . . . . . . . . . . . . 13  |-  ( [. A  /  a ]. ( X F a  /\  -.  a ( t+ `  F ) B )  <->  ( [. A  /  a ]. X F a  /\  [. A  /  a ].  -.  a ( t+ `  F ) B ) )
14 sbcbr2g 4477 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  ( [. A  /  a ]. X F a  <->  X F [_ A  /  a ]_ a ) )
15 csbvarg 3821 . . . . . . . . . . . . . . . 16  |-  ( A  e.  _V  ->  [_ A  /  a ]_ a  =  A )
1615breq2d 4433 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  ( X F [_ A  / 
a ]_ a  <->  X F A ) )
1714, 16bitrd 257 . . . . . . . . . . . . . 14  |-  ( A  e.  _V  ->  ( [. A  /  a ]. X F a  <->  X F A ) )
18 sbcng 3341 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  ( [. A  /  a ].  -.  a ( t+ `  F ) B  <->  -.  [. A  / 
a ]. a ( t+ `  F ) B ) )
19 sbcbr1g 4476 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  _V  ->  ( [. A  /  a ]. a ( t+ `  F ) B  <->  [_ A  /  a ]_ a ( t+ `  F ) B ) )
2015breq1d 4431 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  _V  ->  ( [_ A  /  a ]_ a ( t+ `  F ) B  <-> 
A ( t+ `  F ) B ) )
2119, 20bitrd 257 . . . . . . . . . . . . . . . 16  |-  ( A  e.  _V  ->  ( [. A  /  a ]. a ( t+ `  F ) B  <-> 
A ( t+ `  F ) B ) )
2221notbid 296 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  ->  ( -.  [. A  /  a ]. a ( t+ `  F ) B  <->  -.  A ( t+ `  F ) B ) )
2318, 22bitrd 257 . . . . . . . . . . . . . 14  |-  ( A  e.  _V  ->  ( [. A  /  a ].  -.  a ( t+ `  F ) B  <->  -.  A (
t+ `  F
) B ) )
2417, 23anbi12d 716 . . . . . . . . . . . . 13  |-  ( A  e.  _V  ->  (
( [. A  /  a ]. X F a  /\  [. A  /  a ].  -.  a ( t+ `  F ) B )  <->  ( X F A  /\  -.  A
( t+ `  F ) B ) ) )
2513, 24syl5bb 261 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  ( [. A  /  a ]. ( X F a  /\  -.  a ( t+ `  F
) B )  <->  ( X F A  /\  -.  A
( t+ `  F ) B ) ) )
2612, 25syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( [. A  / 
a ]. ( X F a  /\  -.  a
( t+ `  F ) B )  <-> 
( X F A  /\  -.  A ( t+ `  F
) B ) ) )
27 spesbc 3382 . . . . . . . . . . 11  |-  ( [. A  /  a ]. ( X F a  /\  -.  a ( t+ `  F ) B )  ->  E. a
( X F a  /\  -.  a ( t+ `  F
) B ) )
2826, 27syl6bir 233 . . . . . . . . . 10  |-  ( ph  ->  ( ( X F A  /\  -.  A
( t+ `  F ) B )  ->  E. a ( X F a  /\  -.  a ( t+ `  F ) B ) ) )
298, 28mpand 680 . . . . . . . . 9  |-  ( ph  ->  ( -.  A ( t+ `  F
) B  ->  E. a
( X F a  /\  -.  a ( t+ `  F
) B ) ) )
30 eupicka 2334 . . . . . . . . 9  |-  ( ( E! a  X F a  /\  E. a
( X F a  /\  -.  a ( t+ `  F
) B ) )  ->  A. a ( X F a  ->  -.  a ( t+ `  F ) B ) )
3110, 29, 30syl6an 548 . . . . . . . 8  |-  ( ph  ->  ( -.  A ( t+ `  F
) B  ->  A. a
( X F a  ->  -.  a (
t+ `  F
) B ) ) )
32 frege124d.f . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  _V )
33 funrel 5616 . . . . . . . . . . . . . 14  |-  ( Fun 
F  ->  Rel  F )
342, 33syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  Rel  F )
35 reltrclfv 13075 . . . . . . . . . . . . 13  |-  ( ( F  e.  _V  /\  Rel  F )  ->  Rel  ( t+ `  F ) )
3632, 34, 35syl2anc 666 . . . . . . . . . . . 12  |-  ( ph  ->  Rel  ( t+ `  F ) )
37 brrelex2 4891 . . . . . . . . . . . 12  |-  ( ( Rel  ( t+ `  F )  /\  X ( t+ `  F ) B )  ->  B  e.  _V )
3836, 3, 37syl2anc 666 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  _V )
39 brcog 5018 . . . . . . . . . . 11  |-  ( ( X  e.  dom  F  /\  B  e.  _V )  ->  ( X ( ( t+ `  F )  o.  F
) B  <->  E. a
( X F a  /\  a ( t+ `  F ) B ) ) )
405, 38, 39syl2anc 666 . . . . . . . . . 10  |-  ( ph  ->  ( X ( ( t+ `  F
)  o.  F ) B  <->  E. a ( X F a  /\  a
( t+ `  F ) B ) ) )
4140notbid 296 . . . . . . . . 9  |-  ( ph  ->  ( -.  X ( ( t+ `  F )  o.  F
) B  <->  -.  E. a
( X F a  /\  a ( t+ `  F ) B ) ) )
42 alinexa 1709 . . . . . . . . 9  |-  ( A. a ( X F a  ->  -.  a
( t+ `  F ) B )  <->  -.  E. a ( X F a  /\  a
( t+ `  F ) B ) )
4341, 42syl6rbbr 268 . . . . . . . 8  |-  ( ph  ->  ( A. a ( X F a  ->  -.  a ( t+ `  F ) B )  <->  -.  X (
( t+ `  F )  o.  F
) B ) )
4431, 43sylibd 218 . . . . . . 7  |-  ( ph  ->  ( -.  A ( t+ `  F
) B  ->  -.  X ( ( t+ `  F )  o.  F ) B ) )
45 brdif 4472 . . . . . . . 8  |-  ( X ( ( t+ `  F )  \ 
( ( t+ `  F )  o.  F ) ) B  <-> 
( X ( t+ `  F ) B  /\  -.  X
( ( t+ `  F )  o.  F ) B ) )
4645simplbi2 630 . . . . . . 7  |-  ( X ( t+ `  F ) B  -> 
( -.  X ( ( t+ `  F )  o.  F
) B  ->  X
( ( t+ `  F )  \ 
( ( t+ `  F )  o.  F ) ) B ) )
473, 44, 46sylsyld 59 . . . . . 6  |-  ( ph  ->  ( -.  A ( t+ `  F
) B  ->  X
( ( t+ `  F )  \ 
( ( t+ `  F )  o.  F ) ) B ) )
48 trclfvdecomr 36186 . . . . . . . . . . 11  |-  ( F  e.  _V  ->  (
t+ `  F
)  =  ( F  u.  ( ( t+ `  F )  o.  F ) ) )
4932, 48syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( t+ `  F )  =  ( F  u.  ( ( t+ `  F
)  o.  F ) ) )
50 uncom 3611 . . . . . . . . . 10  |-  ( F  u.  ( ( t+ `  F )  o.  F ) )  =  ( ( ( t+ `  F
)  o.  F )  u.  F )
5149, 50syl6eq 2480 . . . . . . . . 9  |-  ( ph  ->  ( t+ `  F )  =  ( ( ( t+ `  F )  o.  F )  u.  F
) )
52 eqimss 3517 . . . . . . . . 9  |-  ( ( t+ `  F
)  =  ( ( ( t+ `  F )  o.  F
)  u.  F )  ->  ( t+ `  F )  C_  ( ( ( t+ `  F )  o.  F )  u.  F ) )
5351, 52syl 17 . . . . . . . 8  |-  ( ph  ->  ( t+ `  F )  C_  (
( ( t+ `  F )  o.  F )  u.  F
) )
54 ssundif 3880 . . . . . . . 8  |-  ( ( t+ `  F
)  C_  ( (
( t+ `  F )  o.  F
)  u.  F )  <-> 
( ( t+ `  F )  \ 
( ( t+ `  F )  o.  F ) )  C_  F )
5553, 54sylib 200 . . . . . . 7  |-  ( ph  ->  ( ( t+ `  F )  \ 
( ( t+ `  F )  o.  F ) )  C_  F )
5655ssbrd 4463 . . . . . 6  |-  ( ph  ->  ( X ( ( t+ `  F
)  \  ( (
t+ `  F
)  o.  F ) ) B  ->  X F B ) )
5747, 56syld 46 . . . . 5  |-  ( ph  ->  ( -.  A ( t+ `  F
) B  ->  X F B ) )
58 funbrfv 5917 . . . . 5  |-  ( Fun 
F  ->  ( X F B  ->  ( F `
 X )  =  B ) )
592, 57, 58sylsyld 59 . . . 4  |-  ( ph  ->  ( -.  A ( t+ `  F
) B  ->  ( F `  X )  =  B ) )
60 eqcom 2432 . . . 4  |-  ( ( F `  X )  =  B  <->  B  =  ( F `  X ) )
6159, 60syl6ib 230 . . 3  |-  ( ph  ->  ( -.  A ( t+ `  F
) B  ->  B  =  ( F `  X ) ) )
62 eqtr3 2451 . . 3  |-  ( ( A  =  ( F `
 X )  /\  B  =  ( F `  X ) )  ->  A  =  B )
631, 61, 62syl6an 548 . 2  |-  ( ph  ->  ( -.  A ( t+ `  F
) B  ->  A  =  B ) )
6463orrd 380 1  |-  ( ph  ->  ( A ( t+ `  F ) B  \/  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371   A.wal 1436    = wceq 1438   E.wex 1660    e. wcel 1869   E!weu 2266   _Vcvv 3082   [.wsbc 3300   [_csb 3396    \ cdif 3434    u. cun 3435    C_ wss 3437   class class class wbr 4421   dom cdm 4851    o. ccom 4855   Rel wrel 4856   Fun wfun 5593   ` cfv 5599   t+ctcl 13043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-seq 12215  df-trcl 13045  df-relexp 13078
This theorem is referenced by:  frege126d  36220
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