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Theorem frege131d 36427
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 36661. (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 13149 . . . . 5  |-  ( F  e.  _V  ->  F  C_  ( t+ `  F ) )
3 imass1 5209 . . . . 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 3589 . . . . 5  |-  ( ( t+ `  F
) " U ) 
C_  ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) )
6 ssun2 3589 . . . . 5  |-  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) )  C_  ( U  u.  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) ) )
75, 6sstri 3427 . . . 4  |-  ( ( t+ `  F
) " U ) 
C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) )
84, 7syl6ss 3430 . . 3  |-  ( ph  ->  ( F " U
)  C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) )
9 trclfvdecomr 36391 . . . . . . . . . . . 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 5015 . . . . . . . . . 10  |-  ( ph  ->  `' ( t+ `  F )  =  `' ( F  u.  ( ( t+ `  F )  o.  F ) ) )
12 cnvun 5247 . . . . . . . . . . 11  |-  `' ( F  u.  ( ( t+ `  F
)  o.  F ) )  =  ( `' F  u.  `' ( ( t+ `  F )  o.  F
) )
13 cnvco 5025 . . . . . . . . . . . 12  |-  `' ( ( t+ `  F )  o.  F
)  =  ( `' F  o.  `' ( t+ `  F
) )
1413uneq2i 3576 . . . . . . . . . . 11  |-  ( `' F  u.  `' ( ( t+ `  F )  o.  F
) )  =  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) )
1512, 14eqtri 2493 . . . . . . . . . 10  |-  `' ( F  u.  ( ( t+ `  F
)  o.  F ) )  =  ( `' F  u.  ( `' F  o.  `' ( t+ `  F
) ) )
1611, 15syl6eq 2521 . . . . . . . . 9  |-  ( ph  ->  `' ( t+ `  F )  =  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) ) )
1716coeq2d 5002 . . . . . . . 8  |-  ( ph  ->  ( F  o.  `' ( t+ `  F ) )  =  ( F  o.  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) ) ) )
18 coundi 5343 . . . . . . . . 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 5852 . . . . . . . . . . 11  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
2119, 20syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
22 coass 5361 . . . . . . . . . . . 12  |-  ( ( F  o.  `' F
)  o.  `' ( t+ `  F
) )  =  ( F  o.  ( `' F  o.  `' ( t+ `  F
) ) )
2322eqcomi 2480 . . . . . . . . . . 11  |-  ( F  o.  ( `' F  o.  `' ( t+ `  F ) ) )  =  ( ( F  o.  `' F
)  o.  `' ( t+ `  F
) )
2421coeq1d 5001 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  o.  `' F )  o.  `' ( t+ `  F ) )  =  ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) )
2523, 24syl5eq 2517 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  ( `' F  o.  `' ( t+ `  F ) ) )  =  ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) )
2621, 25uneq12d 3580 . . . . . . . . 9  |-  ( ph  ->  ( ( F  o.  `' F )  u.  ( F  o.  ( `' F  o.  `' (
t+ `  F
) ) ) )  =  ( (  _I  |`  ran  F )  u.  ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) ) )
2718, 26syl5eq 2517 . . . . . . . 8  |-  ( ph  ->  ( F  o.  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) ) )  =  ( (  _I  |`  ran  F )  u.  ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) ) )
2817, 27eqtrd 2505 . . . . . . 7  |-  ( ph  ->  ( F  o.  `' ( t+ `  F ) )  =  ( (  _I  |`  ran  F
)  u.  ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) ) )
2928imaeq1d 5173 . . . . . 6  |-  ( ph  ->  ( ( F  o.  `' ( t+ `  F ) )
" U )  =  ( ( (  _I  |`  ran  F )  u.  ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) ) " U ) )
30 imaundir 5255 . . . . . 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 2521 . . . . 5  |-  ( ph  ->  ( ( F  o.  `' ( t+ `  F ) )
" U )  =  ( ( (  _I  |`  ran  F ) " U )  u.  (
( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) " U
) ) )
32 resss 5134 . . . . . . . . 9  |-  (  _I  |`  ran  F )  C_  _I
33 imass1 5209 . . . . . . . . 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 5186 . . . . . . . 8  |-  (  _I  " U )  =  U
3634, 35sseqtri 3450 . . . . . . 7  |-  ( (  _I  |`  ran  F )
" U )  C_  U
37 imaco 5347 . . . . . . . 8  |-  ( ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) " U
)  =  ( (  _I  |`  ran  F )
" ( `' ( t+ `  F
) " U ) )
38 imass1 5209 . . . . . . . . . 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 5186 . . . . . . . . 9  |-  (  _I  " ( `' ( t+ `  F
) " U ) )  =  ( `' ( t+ `  F ) " U
)
4139, 40sseqtri 3450 . . . . . . . 8  |-  ( (  _I  |`  ran  F )
" ( `' ( t+ `  F
) " U ) )  C_  ( `' ( t+ `  F ) " U
)
4237, 41eqsstri 3448 . . . . . . 7  |-  ( ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) " U
)  C_  ( `' ( t+ `  F ) " U
)
43 unss12 3597 . . . . . . 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 686 . . . . . 6  |-  ( ( (  _I  |`  ran  F
) " U )  u.  ( ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) " U ) )  C_  ( U  u.  ( `' ( t+ `  F )
" U ) )
45 ssun1 3588 . . . . . . 7  |-  ( U  u.  ( `' ( t+ `  F
) " U ) )  C_  ( ( U  u.  ( `' ( t+ `  F ) " U
) )  u.  (
( t+ `  F ) " U
) )
46 unass 3582 . . . . . . 7  |-  ( ( U  u.  ( `' ( t+ `  F ) " U
) )  u.  (
( t+ `  F ) " U
) )  =  ( U  u.  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) ) )
4745, 46sseqtri 3450 . . . . . 6  |-  ( U  u.  ( `' ( t+ `  F
) " U ) )  C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) )
4844, 47sstri 3427 . . . . 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 3468 . . . 4  |-  ( ph  ->  ( ( F  o.  `' ( t+ `  F ) )
" U )  C_  ( U  u.  (
( `' ( t+ `  F )
" U )  u.  ( ( t+ `  F ) " U ) ) ) )
50 coss1 4995 . . . . . . . 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 13157 . . . . . . 7  |-  ( ( t+ `  F
)  o.  ( t+ `  F ) )  C_  ( t+ `  F )
5351, 52syl6ss 3430 . . . . . 6  |-  ( ph  ->  ( F  o.  (
t+ `  F
) )  C_  (
t+ `  F
) )
54 imass1 5209 . . . . . 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 3430 . . . 4  |-  ( ph  ->  ( ( F  o.  ( t+ `  F ) ) " U )  C_  ( U  u.  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) ) ) )
5749, 56unssd 3601 . . 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 3601 . 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 5174 . . 3  |-  ( ph  ->  ( F " A
)  =  ( F
" ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) ) )
61 imaundi 5254 . . . 4  |-  ( F
" ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) )  =  ( ( F " U
)  u.  ( F
" ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) ) ) )
62 imaundi 5254 . . . . . 6  |-  ( F
" ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) ) )  =  ( ( F " ( `' ( t+ `  F ) " U ) )  u.  ( F " (
( t+ `  F ) " U
) ) )
63 imaco 5347 . . . . . . . 8  |-  ( ( F  o.  `' ( t+ `  F
) ) " U
)  =  ( F
" ( `' ( t+ `  F
) " U ) )
6463eqcomi 2480 . . . . . . 7  |-  ( F
" ( `' ( t+ `  F
) " U ) )  =  ( ( F  o.  `' ( t+ `  F
) ) " U
)
65 imaco 5347 . . . . . . . 8  |-  ( ( F  o.  ( t+ `  F ) ) " U )  =  ( F "
( ( t+ `  F ) " U ) )
6665eqcomi 2480 . . . . . . 7  |-  ( F
" ( ( t+ `  F )
" U ) )  =  ( ( F  o.  ( t+ `  F ) )
" U )
6764, 66uneq12i 3577 . . . . . 6  |-  ( ( F " ( `' ( t+ `  F ) " U
) )  u.  ( F " ( ( t+ `  F )
" U ) ) )  =  ( ( ( F  o.  `' ( t+ `  F ) ) " U )  u.  (
( F  o.  (
t+ `  F
) ) " U
) )
6862, 67eqtri 2493 . . . . 5  |-  ( F
" ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) ) )  =  ( ( ( F  o.  `' ( t+ `  F ) )
" U )  u.  ( ( F  o.  ( t+ `  F ) ) " U ) )
6968uneq2i 3576 . . . 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 2493 . . 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 2521 . 2  |-  ( ph  ->  ( F " A
)  =  ( ( F " U )  u.  ( ( ( F  o.  `' ( t+ `  F
) ) " U
)  u.  ( ( F  o.  ( t+ `  F ) ) " U ) ) ) )
7258, 71, 593sstr4d 3461 1  |-  ( ph  ->  ( F " A
)  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   _Vcvv 3031    u. cun 3388    C_ wss 3390    _I cid 4749   `'ccnv 4838   ran crn 4840    |` cres 4841   "cima 4842    o. ccom 4843   Fun wfun 5583   ` cfv 5589   t+ctcl 13124
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-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-1st 6812  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-fz 11811  df-seq 12252  df-trcl 13126  df-relexp 13161
This theorem is referenced by:  frege133d  36428
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