frege131d - Mathbox for Richard Penner

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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	⊢ (𝜑 → 𝐹 ∈ V)
frege131d.a	⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
frege131d.fun	⊢ (𝜑 → Fun 𝐹)

**Assertion**
Ref	Expression
frege131d	⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

**Proof of Theorem frege131d**
Step	Hyp	Ref	Expression
1		frege131d.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
2		trclfvlb 13749	. . . . 5 ⊢ (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3		imass1 5500	. . . . 5 ⊢ (𝐹 ⊆ (t+‘𝐹) → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5		ssun2 3777	. . . . 5 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6		ssun2 3777	. . . . 5 ⊢ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
7	5, 6	sstri 3612	. . . 4 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
8	4, 7	syl6ss 3615	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9		trclfvdecomr 38020	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ V → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
10	1, 9	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
11	10	cnveqd 5298	. . . . . . . . . 10 ⊢ (𝜑 → ^◡(t+‘𝐹) = ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12		cnvun 5538	. . . . . . . . . . 11 ⊢ ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ ^◡((t+‘𝐹) ∘ 𝐹))
13		cnvco 5308	. . . . . . . . . . . 12 ⊢ ^◡((t+‘𝐹) ∘ 𝐹) = (^◡𝐹 ∘ ^◡(t+‘𝐹))
14	13	uneq2i 3764	. . . . . . . . . . 11 ⊢ (^◡𝐹 ∪ ^◡((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
15	12, 14	eqtri 2644	. . . . . . . . . 10 ⊢ ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
16	11, 15	syl6eq 2672	. . . . . . . . 9 ⊢ (𝜑 → ^◡(t+‘𝐹) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹))))
17	16	coeq2d 5284	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ ^◡(t+‘𝐹)) = (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))))
18		coundi 5636	. . . . . . . . 9 ⊢ (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = ((𝐹 ∘ ^◡𝐹) ∪ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))))
19		frege131d.fun	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
20		funcocnv2 6161	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))
22		coass 5654	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹)) = (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
23	22	eqcomi 2631	. . . . . . . . . . 11 ⊢ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))) = ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹))
24	21	coeq1d 5283	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)))
25	23, 24	syl5eq 2668	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)))
26	21, 25	uneq12d 3768	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ ^◡𝐹) ∪ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
27	18, 26	syl5eq 2668	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
28	17, 27	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ^◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
29	28	imaeq1d 5465	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))) “ 𝑈))
30		imaundir 5546	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈))
31	29, 30	syl6eq 2672	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)))
32		resss 5422	. . . . . . . . 9 ⊢ ( I ↾ ran 𝐹) ⊆ I
33		imass1 5500	. . . . . . . . 9 ⊢ (( I ↾ ran 𝐹) ⊆ I → (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈))
34	32, 33	ax-mp 5	. . . . . . . 8 ⊢ (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈)
35		imai 5478	. . . . . . . 8 ⊢ ( I “ 𝑈) = 𝑈
36	34, 35	sseqtri 3637	. . . . . . 7 ⊢ (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37		imaco 5640	. . . . . . . 8 ⊢ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ (^◡(t+‘𝐹) “ 𝑈))
38		imass1 5500	. . . . . . . . . 10 ⊢ (( I ↾ ran 𝐹) ⊆ I → (( I ↾ ran 𝐹) “ (^◡(t+‘𝐹) “ 𝑈)) ⊆ ( I “ (^◡(t+‘𝐹) “ 𝑈)))
39	32, 38	ax-mp 5	. . . . . . . . 9 ⊢ (( I ↾ ran 𝐹) “ (^◡(t+‘𝐹) “ 𝑈)) ⊆ ( 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)) ⊆ (^◡(t+‘𝐹) “ 𝑈)
42	37, 41	eqsstri 3635	. . . . . . 7 ⊢ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈) ⊆ (^◡(t+‘𝐹) “ 𝑈)
43		unss12 3785	. . . . . . 7 ⊢ (((( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈 ∧ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈) ⊆ (^◡(t+‘𝐹) “ 𝑈)) → ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)))
44	36, 42, 43	mp2an 708	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈))
45		ssun1 3776	. . . . . . 7 ⊢ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46		unass 3770	. . . . . . 7 ⊢ ((𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
47	45, 46	sseqtri 3637	. . . . . 6 ⊢ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
48	44, 47	sstri 3612	. . . . 5 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
49	31, 48	syl6eqss 3655	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50		coss1 5277	. . . . . . . 8 ⊢ (𝐹 ⊆ (t+‘𝐹) → (𝐹 ∘ (t+‘𝐹)) ⊆ ((t+‘𝐹) ∘ (t+‘𝐹)))
51	1, 2, 50	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ ((t+‘𝐹) ∘ (t+‘𝐹)))
52		trclfvcotrg 13757	. . . . . . 7 ⊢ ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
53	51, 52	syl6ss 3615	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54		imass1 5500	. . . . . 6 ⊢ ((𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹) → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
55	53, 54	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
56	55, 7	syl6ss 3615	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
57	49, 56	unssd 3789	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
58	8, 57	unssd 3789	. 2 ⊢ (𝜑 → ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
59		frege131d.a	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
60	59	imaeq2d 5466	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61		imaundi 5545	. . . 4 ⊢ (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62		imaundi 5545	. . . . . 6 ⊢ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63		imaco 5640	. . . . . . . 8 ⊢ ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = (𝐹 “ (^◡(t+‘𝐹) “ 𝑈))
64	63	eqcomi 2631	. . . . . . 7 ⊢ (𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈)
65		imaco 5640	. . . . . . . 8 ⊢ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
66	65	eqcomi 2631	. . . . . . 7 ⊢ (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
67	64, 66	uneq12i 3765	. . . . . 6 ⊢ ((𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
68	62, 67	eqtri 2644	. . . . 5 ⊢ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
69	68	uneq2i 3764	. . . 4 ⊢ ((𝐹 “ 𝑈) ∪ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
70	61, 69	eqtri 2644	. . 3 ⊢ (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
71	60, 70	syl6eq 2672	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
72	58, 71, 59	3sstr4d 3648	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 I cid 5023 ^◡ccnv 5113 ran crn 5115 ↾ cres 5116 “ cima 5117 ∘ 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

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