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Theorem frege131d 36350
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 36584. (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 13065 . . . . 5  |-  ( F  e.  _V  ->  F  C_  ( t+ `  F ) )
3 imass1 5202 . . . . 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 3597 . . . . 5  |-  ( ( t+ `  F
) " U ) 
C_  ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) )
6 ssun2 3597 . . . . 5  |-  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) )  C_  ( U  u.  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) ) )
75, 6sstri 3440 . . . 4  |-  ( ( t+ `  F
) " U ) 
C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) )
84, 7syl6ss 3443 . . 3  |-  ( ph  ->  ( F " U
)  C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) )
9 trclfvdecomr 36314 . . . . . . . . . . . 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 5009 . . . . . . . . . 10  |-  ( ph  ->  `' ( t+ `  F )  =  `' ( F  u.  ( ( t+ `  F )  o.  F ) ) )
12 cnvun 5240 . . . . . . . . . . 11  |-  `' ( F  u.  ( ( t+ `  F
)  o.  F ) )  =  ( `' F  u.  `' ( ( t+ `  F )  o.  F
) )
13 cnvco 5019 . . . . . . . . . . . 12  |-  `' ( ( t+ `  F )  o.  F
)  =  ( `' F  o.  `' ( t+ `  F
) )
1413uneq2i 3584 . . . . . . . . . . 11  |-  ( `' F  u.  `' ( ( t+ `  F )  o.  F
) )  =  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) )
1512, 14eqtri 2472 . . . . . . . . . 10  |-  `' ( F  u.  ( ( t+ `  F
)  o.  F ) )  =  ( `' F  u.  ( `' F  o.  `' ( t+ `  F
) ) )
1611, 15syl6eq 2500 . . . . . . . . 9  |-  ( ph  ->  `' ( t+ `  F )  =  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) ) )
1716coeq2d 4996 . . . . . . . 8  |-  ( ph  ->  ( F  o.  `' ( t+ `  F ) )  =  ( F  o.  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) ) ) )
18 coundi 5335 . . . . . . . . 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 5836 . . . . . . . . . . 11  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
2119, 20syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
22 coass 5353 . . . . . . . . . . . 12  |-  ( ( F  o.  `' F
)  o.  `' ( t+ `  F
) )  =  ( F  o.  ( `' F  o.  `' ( t+ `  F
) ) )
2322eqcomi 2459 . . . . . . . . . . 11  |-  ( F  o.  ( `' F  o.  `' ( t+ `  F ) ) )  =  ( ( F  o.  `' F
)  o.  `' ( t+ `  F
) )
2421coeq1d 4995 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  o.  `' F )  o.  `' ( t+ `  F ) )  =  ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) )
2523, 24syl5eq 2496 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  ( `' F  o.  `' ( t+ `  F ) ) )  =  ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) )
2621, 25uneq12d 3588 . . . . . . . . 9  |-  ( ph  ->  ( ( F  o.  `' F )  u.  ( F  o.  ( `' F  o.  `' (
t+ `  F
) ) ) )  =  ( (  _I  |`  ran  F )  u.  ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) ) )
2718, 26syl5eq 2496 . . . . . . . 8  |-  ( ph  ->  ( F  o.  ( `' F  u.  ( `' F  o.  `' ( t+ `  F ) ) ) )  =  ( (  _I  |`  ran  F )  u.  ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) ) )
2817, 27eqtrd 2484 . . . . . . 7  |-  ( ph  ->  ( F  o.  `' ( t+ `  F ) )  =  ( (  _I  |`  ran  F
)  u.  ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) ) )
2928imaeq1d 5166 . . . . . 6  |-  ( ph  ->  ( ( F  o.  `' ( t+ `  F ) )
" U )  =  ( ( (  _I  |`  ran  F )  u.  ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) ) " U ) )
30 imaundir 5248 . . . . . 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 2500 . . . . 5  |-  ( ph  ->  ( ( F  o.  `' ( t+ `  F ) )
" U )  =  ( ( (  _I  |`  ran  F ) " U )  u.  (
( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) " U
) ) )
32 resss 5127 . . . . . . . . 9  |-  (  _I  |`  ran  F )  C_  _I
33 imass1 5202 . . . . . . . . 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 5179 . . . . . . . 8  |-  (  _I  " U )  =  U
3634, 35sseqtri 3463 . . . . . . 7  |-  ( (  _I  |`  ran  F )
" U )  C_  U
37 imaco 5339 . . . . . . . 8  |-  ( ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) " U
)  =  ( (  _I  |`  ran  F )
" ( `' ( t+ `  F
) " U ) )
38 imass1 5202 . . . . . . . . . 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 5179 . . . . . . . . 9  |-  (  _I  " ( `' ( t+ `  F
) " U ) )  =  ( `' ( t+ `  F ) " U
)
4139, 40sseqtri 3463 . . . . . . . 8  |-  ( (  _I  |`  ran  F )
" ( `' ( t+ `  F
) " U ) )  C_  ( `' ( t+ `  F ) " U
)
4237, 41eqsstri 3461 . . . . . . 7  |-  ( ( (  _I  |`  ran  F
)  o.  `' ( t+ `  F
) ) " U
)  C_  ( `' ( t+ `  F ) " U
)
43 unss12 3605 . . . . . . 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 677 . . . . . 6  |-  ( ( (  _I  |`  ran  F
) " U )  u.  ( ( (  _I  |`  ran  F )  o.  `' ( t+ `  F ) ) " U ) )  C_  ( U  u.  ( `' ( t+ `  F )
" U ) )
45 ssun1 3596 . . . . . . 7  |-  ( U  u.  ( `' ( t+ `  F
) " U ) )  C_  ( ( U  u.  ( `' ( t+ `  F ) " U
) )  u.  (
( t+ `  F ) " U
) )
46 unass 3590 . . . . . . 7  |-  ( ( U  u.  ( `' ( t+ `  F ) " U
) )  u.  (
( t+ `  F ) " U
) )  =  ( U  u.  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) ) )
4745, 46sseqtri 3463 . . . . . 6  |-  ( U  u.  ( `' ( t+ `  F
) " U ) )  C_  ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) )
4844, 47sstri 3440 . . . . 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 3481 . . . 4  |-  ( ph  ->  ( ( F  o.  `' ( t+ `  F ) )
" U )  C_  ( U  u.  (
( `' ( t+ `  F )
" U )  u.  ( ( t+ `  F ) " U ) ) ) )
50 coss1 4989 . . . . . . . 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 13073 . . . . . . 7  |-  ( ( t+ `  F
)  o.  ( t+ `  F ) )  C_  ( t+ `  F )
5351, 52syl6ss 3443 . . . . . 6  |-  ( ph  ->  ( F  o.  (
t+ `  F
) )  C_  (
t+ `  F
) )
54 imass1 5202 . . . . . 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 3443 . . . 4  |-  ( ph  ->  ( ( F  o.  ( t+ `  F ) ) " U )  C_  ( U  u.  ( ( `' ( t+ `  F ) " U )  u.  (
( t+ `  F ) " U
) ) ) )
5749, 56unssd 3609 . . 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 3609 . 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 5167 . . 3  |-  ( ph  ->  ( F " A
)  =  ( F
" ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) ) )
61 imaundi 5247 . . . 4  |-  ( F
" ( U  u.  ( ( `' ( t+ `  F
) " U )  u.  ( ( t+ `  F )
" U ) ) ) )  =  ( ( F " U
)  u.  ( F
" ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) ) ) )
62 imaundi 5247 . . . . . 6  |-  ( F
" ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) ) )  =  ( ( F " ( `' ( t+ `  F ) " U ) )  u.  ( F " (
( t+ `  F ) " U
) ) )
63 imaco 5339 . . . . . . . 8  |-  ( ( F  o.  `' ( t+ `  F
) ) " U
)  =  ( F
" ( `' ( t+ `  F
) " U ) )
6463eqcomi 2459 . . . . . . 7  |-  ( F
" ( `' ( t+ `  F
) " U ) )  =  ( ( F  o.  `' ( t+ `  F
) ) " U
)
65 imaco 5339 . . . . . . . 8  |-  ( ( F  o.  ( t+ `  F ) ) " U )  =  ( F "
( ( t+ `  F ) " U ) )
6665eqcomi 2459 . . . . . . 7  |-  ( F
" ( ( t+ `  F )
" U ) )  =  ( ( F  o.  ( t+ `  F ) )
" U )
6764, 66uneq12i 3585 . . . . . 6  |-  ( ( F " ( `' ( t+ `  F ) " U
) )  u.  ( F " ( ( t+ `  F )
" U ) ) )  =  ( ( ( F  o.  `' ( t+ `  F ) ) " U )  u.  (
( F  o.  (
t+ `  F
) ) " U
) )
6862, 67eqtri 2472 . . . . 5  |-  ( F
" ( ( `' ( t+ `  F ) " U
)  u.  ( ( t+ `  F
) " U ) ) )  =  ( ( ( F  o.  `' ( t+ `  F ) )
" U )  u.  ( ( F  o.  ( t+ `  F ) ) " U ) )
6968uneq2i 3584 . . . 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 2472 . . 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 2500 . 2  |-  ( ph  ->  ( F " A
)  =  ( ( F " U )  u.  ( ( ( F  o.  `' ( t+ `  F
) ) " U
)  u.  ( ( F  o.  ( t+ `  F ) ) " U ) ) ) )
7258, 71, 593sstr4d 3474 1  |-  ( ph  ->  ( F " A
)  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1443    e. wcel 1886   _Vcvv 3044    u. cun 3401    C_ wss 3403    _I cid 4743   `'ccnv 4832   ran crn 4834    |` cres 4835   "cima 4836    o. ccom 4837   Fun wfun 5575   ` cfv 5581   t+ctcl 13042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-seq 12211  df-trcl 13044  df-relexp 13077
This theorem is referenced by:  frege133d  36351
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