Theorem frege131d 38056

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 38288. (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

Step	Hyp	Ref	Expression
1		frege131d.f	. . . . 5 |- ( ph -> F e. _V )
2		trclfvlb 13749	. . . . 5 |- ( F e. _V -> F C_ ( t+ ` F ) )
3		imass1 5500	. . . . 5 |- ( F C_ ( t+ ` F ) -> ( F " U ) C_ ( ( t+ ` F ) " U ) )
4	1, 2, 3	3syl 18	. . . 4 |- ( ph -> ( F " U ) C_ ( ( t+ ` F ) " U ) )
5		ssun2 3777	. . . . 5 |- ( ( t+ ` F ) " U ) C_ ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) )
6		ssun2 3777	. . . . 5 |- ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) C_ ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) )
7	5, 6	sstri 3612	. . . 4 |- ( ( t+ ` F ) " U ) C_ ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) )
8	4, 7	syl6ss 3615	. . 3 |- ( ph -> ( F " U ) C_ ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) ) )
9		trclfvdecomr 38020	. . . . . . . . . . . 12 |- ( F e. _V -> ( t+ ` F ) = ( F u. ( ( t+ ` F ) o. F ) ) )
10	1, 9	syl 17	. . . . . . . . . . 11 |- ( ph -> ( t+ ` F ) = ( F u. ( ( t+ ` F ) o. F ) ) )
11	10	cnveqd 5298	. . . . . . . . . 10 |- ( ph -> `' ( t+ ` F ) = `' ( F u. ( ( t+ ` F ) o. F ) ) )
12		cnvun 5538	. . . . . . . . . . 11 |- `' ( F u. ( ( t+ ` F ) o. F ) ) = ( `' F u. `' ( ( t+ ` F ) o. F ) )
13		cnvco 5308	. . . . . . . . . . . 12 |- `' ( ( t+ ` F ) o. F ) = ( `' F o. `' ( t+ ` F ) )
14	13	uneq2i 3764	. . . . . . . . . . 11 |- ( `' F u. `' ( ( t+ ` F ) o. F ) ) = ( `' F u. ( `' F o. `' ( t+ ` F ) ) )
15	12, 14	eqtri 2644	. . . . . . . . . 10 |- `' ( F u. ( ( t+ ` F ) o. F ) ) = ( `' F u. ( `' F o. `' ( t+ ` F ) ) )
16	11, 15	syl6eq 2672	. . . . . . . . 9 |- ( ph -> `' ( t+ ` F ) = ( `' F u. ( `' F o. `' ( t+ ` F ) ) ) )
17	16	coeq2d 5284	. . . . . . . 8 |- ( ph -> ( F o. `' ( t+ ` F ) ) = ( F o. ( `' F u. ( `' F o. `' ( t+ ` F ) ) ) ) )
18		coundi 5636	. . . . . . . . 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 6161	. . . . . . . . . . 11 |- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )
21	19, 20	syl 17	. . . . . . . . . 10 |- ( ph -> ( F o. `' F ) = ( _I |` ran F ) )
22		coass 5654	. . . . . . . . . . . 12 |- ( ( F o. `' F ) o. `' ( t+ ` F ) ) = ( F o. ( `' F o. `' ( t+ ` F ) ) )
23	22	eqcomi 2631	. . . . . . . . . . 11 |- ( F o. ( `' F o. `' ( t+ ` F ) ) ) = ( ( F o. `' F ) o. `' ( t+ ` F ) )
24	21	coeq1d 5283	. . . . . . . . . . 11 |- ( ph -> ( ( F o. `' F ) o. `' ( t+ ` F ) ) = ( ( _I |` ran F ) o. `' ( t+ ` F ) ) )
25	23, 24	syl5eq 2668	. . . . . . . . . 10 |- ( ph -> ( F o. ( `' F o. `' ( t+ ` F ) ) ) = ( ( _I |` ran F ) o. `' ( t+ ` F ) ) )
26	21, 25	uneq12d 3768	. . . . . . . . 9 |- ( ph -> ( ( F o. `' F ) u. ( F o. ( `' F o. `' ( t+ ` F ) ) ) ) = ( ( _I |` ran F ) u. ( ( _I |` ran F ) o. `' ( t+ ` F ) ) ) )
27	18, 26	syl5eq 2668	. . . . . . . 8 |- ( ph -> ( F o. ( `' F u. ( `' F o. `' ( t+ ` F ) ) ) ) = ( ( _I |` ran F ) u. ( ( _I |` ran F ) o. `' ( t+ ` F ) ) ) )
28	17, 27	eqtrd 2656	. . . . . . 7 |- ( ph -> ( F o. `' ( t+ ` F ) ) = ( ( _I |` ran F ) u. ( ( _I |` ran F ) o. `' ( t+ ` F ) ) ) )
29	28	imaeq1d 5465	. . . . . 6 |- ( ph -> ( ( F o. `' ( t+ ` F ) ) " U ) = ( ( ( _I |` ran F ) u. ( ( _I |` ran F ) o. `' ( t+ ` F ) ) ) " U ) )
30		imaundir 5546	. . . . . 6 |- ( ( ( _I |` ran F ) u. ( ( _I |` ran F ) o. `' ( t+ ` F ) ) ) " U ) = ( ( ( _I |` ran F ) " U ) u. ( ( ( _I |` ran F ) o. `' ( t+ ` F ) ) " U ) )
31	29, 30	syl6eq 2672	. . . . 5 |- ( ph -> ( ( F o. `' ( t+ ` F ) ) " U ) = ( ( ( _I |` ran F ) " U ) u. ( ( ( _I |` ran F ) o. `' ( t+ ` F ) ) " U ) ) )
32		resss 5422	. . . . . . . . 9 |- ( _I |` ran F ) C_ _I
33		imass1 5500	. . . . . . . . 9 |- ( ( _I |` ran F ) C_ _I -> ( ( _I |` ran F ) " U ) C_ ( _I " U ) )
34	32, 33	ax-mp 5	. . . . . . . 8 |- ( ( _I |` ran F ) " U ) C_ ( _I " U )
35		imai 5478	. . . . . . . 8 |- ( _I " U ) = U
36	34, 35	sseqtri 3637	. . . . . . 7 |- ( ( _I |` ran F ) " U ) C_ U
37		imaco 5640	. . . . . . . 8 |- ( ( ( _I |` ran F ) o. `' ( t+ ` F ) ) " U ) = ( ( _I |` ran F ) " ( `' ( t+ ` F ) " U ) )
38		imass1 5500	. . . . . . . . . 10 |- ( ( _I |` ran F ) C_ _I -> ( ( _I |` ran F ) " ( `' ( t+ ` F ) " U ) ) C_ ( _I " ( `' ( t+ ` F ) " U ) ) )
39	32, 38	ax-mp 5	. . . . . . . . 9 |- ( ( _I |` ran F ) " ( `' ( t+ ` F ) " U ) ) C_ ( _I " ( `' ( t+ ` F ) " U ) )
40		imai 5478	. . . . . . . . 9 |- ( _I " ( `' ( t+ ` F ) " U ) ) = ( `' ( t+ ` F ) " U )
41	39, 40	sseqtri 3637	. . . . . . . 8 |- ( ( _I |` ran F ) " ( `' ( t+ ` F ) " U ) ) C_ ( `' ( t+ ` F ) " U )
42	37, 41	eqsstri 3635	. . . . . . 7 |- ( ( ( _I |` ran F ) o. `' ( t+ ` F ) ) " U ) C_ ( `' ( t+ ` F ) " U )
43		unss12 3785	. . . . . . 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 ) ) )
44	36, 42, 43	mp2an 708	. . . . . 6 |- ( ( ( _I |` ran F ) " U ) u. ( ( ( _I |` ran F ) o. `' ( t+ ` F ) ) " U ) ) C_ ( U u. ( `' ( t+ ` F ) " U ) )
45		ssun1 3776	. . . . . . 7 |- ( U u. ( `' ( t+ ` F ) " U ) ) C_ ( ( U u. ( `' ( t+ ` F ) " U ) ) u. ( ( t+ ` F ) " U ) )
46		unass 3770	. . . . . . 7 |- ( ( U u. ( `' ( t+ ` F ) " U ) ) u. ( ( t+ ` F ) " U ) ) = ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) )
47	45, 46	sseqtri 3637	. . . . . 6 |- ( U u. ( `' ( t+ ` F ) " U ) ) C_ ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) )
48	44, 47	sstri 3612	. . . . 5 |- ( ( ( _I |` ran F ) " U ) u. ( ( ( _I |` ran F ) o. `' ( t+ ` F ) ) " U ) ) C_ ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) )
49	31, 48	syl6eqss 3655	. . . 4 |- ( ph -> ( ( F o. `' ( t+ ` F ) ) " U ) C_ ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) ) )
50		coss1 5277	. . . . . . . 8 |- ( F C_ ( t+ ` F ) -> ( F o. ( t+ ` F ) ) C_ ( ( t+ ` F ) o. ( t+ ` F ) ) )
51	1, 2, 50	3syl 18	. . . . . . 7 |- ( ph -> ( F o. ( t+ ` F ) ) C_ ( ( t+ ` F ) o. ( t+ ` F ) ) )
52		trclfvcotrg 13757	. . . . . . 7 |- ( ( t+ ` F ) o. ( t+ ` F ) ) C_ ( t+ ` F )
53	51, 52	syl6ss 3615	. . . . . 6 |- ( ph -> ( F o. ( t+ ` F ) ) C_ ( t+ ` F ) )
54		imass1 5500	. . . . . 6 |- ( ( F o. ( t+ ` F ) ) C_ ( t+ ` F ) -> ( ( F o. ( t+ ` F ) ) " U ) C_ ( ( t+ ` F ) " U ) )
55	53, 54	syl 17	. . . . 5 |- ( ph -> ( ( F o. ( t+ ` F ) ) " U ) C_ ( ( t+ ` F ) " U ) )
56	55, 7	syl6ss 3615	. . . 4 |- ( ph -> ( ( F o. ( t+ ` F ) ) " U ) C_ ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) ) )
57	49, 56	unssd 3789	. . 3 |- ( ph -> ( ( ( F o. `' ( t+ ` F ) ) " U ) u. ( ( F o. ( t+ ` F ) ) " U ) ) C_ ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) ) )
58	8, 57	unssd 3789	. 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 ) ) ) )
60	59	imaeq2d 5466	. . 3 |- ( ph -> ( F " A ) = ( F " ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) ) ) )
61		imaundi 5545	. . . 4 |- ( F " ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) ) ) = ( ( F " U ) u. ( F " ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) ) )
62		imaundi 5545	. . . . . 6 |- ( F " ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) ) = ( ( F " ( `' ( t+ ` F ) " U ) ) u. ( F " ( ( t+ ` F ) " U ) ) )
63		imaco 5640	. . . . . . . 8 |- ( ( F o. `' ( t+ ` F ) ) " U ) = ( F " ( `' ( t+ ` F ) " U ) )
64	63	eqcomi 2631	. . . . . . 7 |- ( F " ( `' ( t+ ` F ) " U ) ) = ( ( F o. `' ( t+ ` F ) ) " U )
65		imaco 5640	. . . . . . . 8 |- ( ( F o. ( t+ ` F ) ) " U ) = ( F " ( ( t+ ` F ) " U ) )
66	65	eqcomi 2631	. . . . . . 7 |- ( F " ( ( t+ ` F ) " U ) ) = ( ( F o. ( t+ ` F ) ) " U )
67	64, 66	uneq12i 3765	. . . . . 6 |- ( ( F " ( `' ( t+ ` F ) " U ) ) u. ( F " ( ( t+ ` F ) " U ) ) ) = ( ( ( F o. `' ( t+ ` F ) ) " U ) u. ( ( F o. ( t+ ` F ) ) " U ) )
68	62, 67	eqtri 2644	. . . . 5 |- ( F " ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) ) = ( ( ( F o. `' ( t+ ` F ) ) " U ) u. ( ( F o. ( t+ ` F ) ) " U ) )
69	68	uneq2i 3764	. . . 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 ) ) )
70	61, 69	eqtri 2644	. . 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 ) ) )
71	60, 70	syl6eq 2672	. 2 |- ( ph -> ( F " A ) = ( ( F " U ) u. ( ( ( F o. `' ( t+ ` F ) ) " U ) u. ( ( F o. ( t+ ` F ) ) " U ) ) ) )
72	58, 71, 59	3sstr4d 3648	1 |- ( ph -> ( F " A ) C_ A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   _I cid 5023   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   ` cfv 5888   t+ctcl 13724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-trcl 13726  df-relexp 13761

This theorem is referenced by:  frege133d  38057