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Theorem frege131d 36210
Description: If  F is a function and  A contains all elements of  U and all elements before or after those elements of  U in the transitive closure of  F, then the image under  F of  A is a subclass of  A. Similar to Proposition 131 of [Frege1879] p. 85. Compare with frege131 36442. (Contributed by RP, 17-Jul-2020.)
Hypotheses
Ref Expression
frege131d.f  |-  ( ph  ->  F  e.  _V )
frege131d.a  |-  ( ph  ->  A  =  ( U  u.  ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) ) ) )
frege131d.fun  |-  ( ph  ->  Fun  F )
Assertion
Ref Expression
frege131d  |-  ( ph  ->  ( F " A
)  C_  A )

Proof of Theorem frege131d
StepHypRef Expression
1 frege131d.f . . . . 5  |-  ( ph  ->  F  e.  _V )
2 trclfvlb 13051 . . . . 5  |-  ( F  e.  _V  ->  F  C_  ( t+ `  F ) )
3 imass1 5215 . . . . 5  |-  ( F 
C_  ( t+ `  F )  -> 
( F " U
)  C_  ( (
t+ `  F
) " U ) )
41, 2, 33syl 18 . . . 4  |-  ( ph  ->  ( F " U
)  C_  ( (
t+ `  F
) " U ) )
5 ssun2 3627 . . . . 5  |-  ( ( t+ `  F
) " U ) 
C_  ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) )
6 ssun2 3627 . . . . 5  |-  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) )  C_  ( U  u.  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) ) )
75, 6sstri 3470 . . . 4  |-  ( ( t+ `  F
) " U ) 
C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) )
84, 7syl6ss 3473 . . 3  |-  ( ph  ->  ( F " U
)  C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) )
9 trclfvdecomr 36174 . . . . . . . . . . . 12  |-  ( F  e.  _V  ->  (
t+ `  F
)  =  ( F  u.  ( ( t+ `  F )  o.  F ) ) )
101, 9syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( t+ `  F )  =  ( F  u.  ( ( t+ `  F
)  o.  F ) ) )
1110cnveqd 5022 . . . . . . . . . 10  |-  ( ph  ->  `' ( t+ `  F )  =  `' ( F  u.  ( ( t+ `  F )  o.  F ) ) )
12 cnvun 5253 . . . . . . . . . . 11  |-  `' ( F  u.  ( ( t+ `  F
)  o.  F ) )  =  ( `' F  u.  `' ( ( t+ `  F )  o.  F
) )
13 cnvco 5032 . . . . . . . . . . . 12  |-  `' ( ( t+ `  F )  o.  F
)  =  ( `' F  o.  `' ( t+ `  F
) )
1413uneq2i 3614 . . . . . . . . . . 11  |-  ( `' F  u.  `' ( ( t+ `  F )  o.  F
) )  =  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) )
1512, 14eqtri 2449 . . . . . . . . . 10  |-  `' ( F  u.  ( ( t+ `  F
)  o.  F ) )  =  ( `' F  u.  ( `' F  o.  `' ( t+ `  F
) ) )
1611, 15syl6eq 2477 . . . . . . . . 9  |-  ( ph  ->  `' ( t+ `  F )  =  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) ) )
1716coeq2d 5009 . . . . . . . 8  |-  ( ph  ->  ( F  o.  `' ( t+ `  F ) )  =  ( F  o.  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) ) ) )
18 coundi 5348 . . . . . . . . 9  |-  ( F  o.  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) ) )  =  ( ( F  o.  `' F )  u.  ( F  o.  ( `' F  o.  `' (
t+ `  F
) ) ) )
19 frege131d.fun . . . . . . . . . . 11  |-  ( ph  ->  Fun  F )
20 funcocnv2 5847 . . . . . . . . . . 11  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
2119, 20syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
22 coass 5366 . . . . . . . . . . . 12  |-  ( ( F  o.  `' F
)  o.  `' ( t+ `  F
) )  =  ( F  o.  ( `' F  o.  `' ( t+ `  F
) ) )
2322eqcomi 2433 . . . . . . . . . . 11  |-  ( F  o.  ( `' F  o.  `' ( t+ `  F ) ) )  =  ( ( F  o.  `' F
)  o.  `' ( t+ `  F
) )
2421coeq1d 5008 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  o.  `' F )  o.  `' ( t+ `  F ) )  =  ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) )
2523, 24syl5eq 2473 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  ( `' F  o.  `' ( t+ `  F ) ) )  =  ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) )
2621, 25uneq12d 3618 . . . . . . . . 9  |-  ( ph  ->  ( ( F  o.  `' F )  u.  ( F  o.  ( `' F  o.  `' (
t+ `  F
) ) ) )  =  ( (  _I  |`  ran  F )  u.  ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) ) )
2718, 26syl5eq 2473 . . . . . . . 8  |-  ( ph  ->  ( F  o.  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) ) )  =  ( (  _I  |`  ran  F )  u.  ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) ) )
2817, 27eqtrd 2461 . . . . . . 7  |-  ( ph  ->  ( F  o.  `' ( t+ `  F ) )  =  ( (  _I  |`  ran  F
)  u.  ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) ) )
2928imaeq1d 5179 . . . . . 6  |-  ( ph  ->  ( ( F  o.  `' ( t+ `  F ) )
" U )  =  ( ( (  _I  |`  ran  F )  u.  ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) ) " U ) )
30 imaundir 5261 . . . . . 6  |-  ( ( (  _I  |`  ran  F
)  u.  ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) ) " U
)  =  ( ( (  _I  |`  ran  F
) " U )  u.  ( ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) " U ) )
3129, 30syl6eq 2477 . . . . 5  |-  ( ph  ->  ( ( F  o.  `' ( t+ `  F ) )
" U )  =  ( ( (  _I  |`  ran  F ) " U )  u.  (
( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) " U
) ) )
32 resss 5140 . . . . . . . . 9  |-  (  _I  |`  ran  F )  C_  _I
33 imass1 5215 . . . . . . . . 9  |-  ( (  _I  |`  ran  F ) 
C_  _I  ->  ( (  _I  |`  ran  F )
" U )  C_  (  _I  " U ) )
3432, 33ax-mp 5 . . . . . . . 8  |-  ( (  _I  |`  ran  F )
" U )  C_  (  _I  " U )
35 imai 5192 . . . . . . . 8  |-  (  _I  " U )  =  U
3634, 35sseqtri 3493 . . . . . . 7  |-  ( (  _I  |`  ran  F )
" U )  C_  U
37 imaco 5352 . . . . . . . 8  |-  ( ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) " U
)  =  ( (  _I  |`  ran  F )
" ( `' ( t+ `  F
) " U ) )
38 imass1 5215 . . . . . . . . . 10  |-  ( (  _I  |`  ran  F ) 
C_  _I  ->  ( (  _I  |`  ran  F )
" ( `' ( t+ `  F
) " U ) )  C_  (  _I  " ( `' ( t+ `  F )
" U ) ) )
3932, 38ax-mp 5 . . . . . . . . 9  |-  ( (  _I  |`  ran  F )
" ( `' ( t+ `  F
) " U ) )  C_  (  _I  " ( `' ( t+ `  F )
" U ) )
40 imai 5192 . . . . . . . . 9  |-  (  _I  " ( `' ( t+ `  F
) " U ) )  =  ( `' ( t+ `  F ) " U
)
4139, 40sseqtri 3493 . . . . . . . 8  |-  ( (  _I  |`  ran  F )
" ( `' ( t+ `  F
) " U ) )  C_  ( `' ( t+ `  F ) " U
)
4237, 41eqsstri 3491 . . . . . . 7  |-  ( ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) " U
)  C_  ( `' ( t+ `  F ) " U
)
43 unss12 3635 . . . . . . 7  |-  ( ( ( (  _I  |`  ran  F
) " U ) 
C_  U  /\  (
( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) " U
)  C_  ( `' ( t+ `  F ) " U
) )  ->  (
( (  _I  |`  ran  F
) " U )  u.  ( ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) " U ) )  C_  ( U  u.  ( `' ( t+ `  F )
" U ) ) )
4436, 42, 43mp2an 676 . . . . . 6  |-  ( ( (  _I  |`  ran  F
) " U )  u.  ( ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) " U ) )  C_  ( U  u.  ( `' ( t+ `  F )
" U ) )
45 ssun1 3626 . . . . . . 7  |-  ( U  u.  ( `' ( t+ `  F
) " U ) )  C_  ( ( U  u.  ( `' ( t+ `  F ) " U
) )  u.  (
( t+ `  F ) " U
) )
46 unass 3620 . . . . . . 7  |-  ( ( U  u.  ( `' ( t+ `  F ) " U
) )  u.  (
( t+ `  F ) " U
) )  =  ( U  u.  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) ) )
4745, 46sseqtri 3493 . . . . . 6  |-  ( U  u.  ( `' ( t+ `  F
) " U ) )  C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) )
4844, 47sstri 3470 . . . . 5  |-  ( ( (  _I  |`  ran  F
) " U )  u.  ( ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) " U ) )  C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) )
4931, 48syl6eqss 3511 . . . 4  |-  ( ph  ->  ( ( F  o.  `' ( t+ `  F ) )
" U )  C_  ( U  u.  (
( `' ( t+ `  F )
" U )  u.  ( ( t+ `  F ) " U ) ) ) )
50 coss1 5002 . . . . . . . 8  |-  ( F 
C_  ( t+ `  F )  -> 
( F  o.  (
t+ `  F
) )  C_  (
( t+ `  F )  o.  (
t+ `  F
) ) )
511, 2, 503syl 18 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
t+ `  F
) )  C_  (
( t+ `  F )  o.  (
t+ `  F
) ) )
52 trclfvcotrg 13059 . . . . . . 7  |-  ( ( t+ `  F
)  o.  ( t+ `  F ) )  C_  ( t+ `  F )
5351, 52syl6ss 3473 . . . . . 6  |-  ( ph  ->  ( F  o.  (
t+ `  F
) )  C_  (
t+ `  F
) )
54 imass1 5215 . . . . . 6  |-  ( ( F  o.  ( t+ `  F ) )  C_  ( t+ `  F )  ->  ( ( F  o.  ( t+ `  F ) )
" U )  C_  ( ( t+ `  F ) " U ) )
5553, 54syl 17 . . . . 5  |-  ( ph  ->  ( ( F  o.  ( t+ `  F ) ) " U )  C_  (
( t+ `  F ) " U
) )
5655, 7syl6ss 3473 . . . 4  |-  ( ph  ->  ( ( F  o.  ( t+ `  F ) ) " U )  C_  ( U  u.  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) ) ) )
5749, 56unssd 3639 . . 3  |-  ( ph  ->  ( ( ( F  o.  `' ( t+ `  F ) ) " U )  u.  ( ( F  o.  ( t+ `  F ) )
" U ) ) 
C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) )
588, 57unssd 3639 . 2  |-  ( ph  ->  ( ( F " U )  u.  (
( ( F  o.  `' ( t+ `  F ) )
" U )  u.  ( ( F  o.  ( t+ `  F ) ) " U ) ) ) 
C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) )
59 frege131d.a . . . 4  |-  ( ph  ->  A  =  ( U  u.  ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) ) ) )
6059imaeq2d 5180 . . 3  |-  ( ph  ->  ( F " A
)  =  ( F
" ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) ) )
61 imaundi 5260 . . . 4  |-  ( F
" ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) )  =  ( ( F " U
)  u.  ( F
" ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) ) ) )
62 imaundi 5260 . . . . . 6  |-  ( F
" ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) ) )  =  ( ( F " ( `' ( t+ `  F ) " U ) )  u.  ( F " (
( t+ `  F ) " U
) ) )
63 imaco 5352 . . . . . . . 8  |-  ( ( F  o.  `' ( t+ `  F
) ) " U
)  =  ( F
" ( `' ( t+ `  F
) " U ) )
6463eqcomi 2433 . . . . . . 7  |-  ( F
" ( `' ( t+ `  F
) " U ) )  =  ( ( F  o.  `' ( t+ `  F
) ) " U
)
65 imaco 5352 . . . . . . . 8  |-  ( ( F  o.  ( t+ `  F ) ) " U )  =  ( F "
( ( t+ `  F ) " U ) )
6665eqcomi 2433 . . . . . . 7  |-  ( F
" ( ( t+ `  F )
" U ) )  =  ( ( F  o.  ( t+ `  F ) )
" U )
6764, 66uneq12i 3615 . . . . . 6  |-  ( ( F " ( `' ( t+ `  F ) " U
) )  u.  ( F " ( ( t+ `  F )
" U ) ) )  =  ( ( ( F  o.  `' ( t+ `  F ) ) " U )  u.  (
( F  o.  (
t+ `  F
) ) " U
) )
6862, 67eqtri 2449 . . . . 5  |-  ( F
" ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) ) )  =  ( ( ( F  o.  `' ( t+ `  F ) )
" U )  u.  ( ( F  o.  ( t+ `  F ) ) " U ) )
6968uneq2i 3614 . . . 4  |-  ( ( F " U )  u.  ( F "
( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) )  =  ( ( F " U
)  u.  ( ( ( F  o.  `' ( t+ `  F ) ) " U )  u.  (
( F  o.  (
t+ `  F
) ) " U
) ) )
7061, 69eqtri 2449 . . 3  |-  ( F
" ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) )  =  ( ( F " U
)  u.  ( ( ( F  o.  `' ( t+ `  F ) ) " U )  u.  (
( F  o.  (
t+ `  F
) ) " U
) ) )
7160, 70syl6eq 2477 . 2  |-  ( ph  ->  ( F " A
)  =  ( ( F " U )  u.  ( ( ( F  o.  `' ( t+ `  F
) ) " U
)  u.  ( ( F  o.  ( t+ `  F ) ) " U ) ) ) )
7258, 71, 593sstr4d 3504 1  |-  ( ph  ->  ( F " A
)  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   _Vcvv 3078    u. cun 3431    C_ wss 3433    _I cid 4756   `'ccnv 4845   ran crn 4847    |` cres 4848   "cima 4849    o. ccom 4850   Fun wfun 5587   ` cfv 5593   t+ctcl 13028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-cnex 9591  ax-resscn 9592  ax-1cn 9593  ax-icn 9594  ax-addcl 9595  ax-addrcl 9596  ax-mulcl 9597  ax-mulrcl 9598  ax-mulcom 9599  ax-addass 9600  ax-mulass 9601  ax-distr 9602  ax-i2m1 9603  ax-1ne0 9604  ax-1rid 9605  ax-rnegex 9606  ax-rrecex 9607  ax-cnre 9608  ax-pre-lttri 9609  ax-pre-lttrn 9610  ax-pre-ltadd 9611  ax-pre-mulgt0 9612
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-tr 4513  df-eprel 4757  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-pred 5391  df-ord 5437  df-on 5438  df-lim 5439  df-suc 5440  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6259  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-om 6699  df-1st 6799  df-2nd 6800  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9673  df-mnf 9674  df-xr 9675  df-ltxr 9676  df-le 9677  df-sub 9858  df-neg 9859  df-nn 10606  df-2 10664  df-n0 10866  df-z 10934  df-uz 11156  df-fz 11779  df-seq 12207  df-trcl 13030  df-relexp 13063
This theorem is referenced by:  frege133d  36211
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