frege131d - Mathbox for Richard Penner

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Theorem frege131d 40465

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 40695. (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 14359	. . . . 5 ⊢ (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3		imass1 5931	. . . . 5 ⊢ (𝐹 ⊆ (t+‘𝐹) → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5		ssun2 4100	. . . . 5 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6		ssun2 4100	. . . . 5 ⊢ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
7	5, 6	sstri 3924	. . . 4 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
8	4, 7	sstrdi 3927	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9		trclfvdecomr 40429	. . . . . . . . . . . 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 5710	. . . . . . . . . 10 ⊢ (𝜑 → ^◡(t+‘𝐹) = ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12		cnvun 5968	. . . . . . . . . . 11 ⊢ ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ ^◡((t+‘𝐹) ∘ 𝐹))
13		cnvco 5720	. . . . . . . . . . . 12 ⊢ ^◡((t+‘𝐹) ∘ 𝐹) = (^◡𝐹 ∘ ^◡(t+‘𝐹))
14	13	uneq2i 4087	. . . . . . . . . . 11 ⊢ (^◡𝐹 ∪ ^◡((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
15	12, 14	eqtri 2821	. . . . . . . . . 10 ⊢ ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
16	11, 15	eqtrdi 2849	. . . . . . . . 9 ⊢ (𝜑 → ^◡(t+‘𝐹) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹))))
17	16	coeq2d 5697	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ ^◡(t+‘𝐹)) = (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))))
18		coundi 6067	. . . . . . . . 9 ⊢ (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = ((𝐹 ∘ ^◡𝐹) ∪ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))))
19		frege131d.fun	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
20		funcocnv2 6614	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))
22		coass 6085	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹)) = (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
23	22	eqcomi 2807	. . . . . . . . . . 11 ⊢ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))) = ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹))
24	21	coeq1d 5696	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)))
25	23, 24	syl5eq 2845	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)))
26	21, 25	uneq12d 4091	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ ^◡𝐹) ∪ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
27	18, 26	syl5eq 2845	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
28	17, 27	eqtrd 2833	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ^◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
29	28	imaeq1d 5895	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))) “ 𝑈))
30		imaundir 5976	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈))
31	29, 30	eqtrdi 2849	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)))
32		resss 5843	. . . . . . . . 9 ⊢ ( I ↾ ran 𝐹) ⊆ I
33		imass1 5931	. . . . . . . . 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 5909	. . . . . . . 8 ⊢ ( I “ 𝑈) = 𝑈
36	34, 35	sseqtri 3951	. . . . . . 7 ⊢ (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37		imaco 6071	. . . . . . . 8 ⊢ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ (^◡(t+‘𝐹) “ 𝑈))
38		imass1 5931	. . . . . . . . . 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 5909	. . . . . . . . 9 ⊢ ( I “ (^◡(t+‘𝐹) “ 𝑈)) = (^◡(t+‘𝐹) “ 𝑈)
41	39, 40	sseqtri 3951	. . . . . . . 8 ⊢ (( I ↾ ran 𝐹) “ (^◡(t+‘𝐹) “ 𝑈)) ⊆ (^◡(t+‘𝐹) “ 𝑈)
42	37, 41	eqsstri 3949	. . . . . . 7 ⊢ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈) ⊆ (^◡(t+‘𝐹) “ 𝑈)
43		unss12 4109	. . . . . . 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 691	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈))
45		ssun1 4099	. . . . . . 7 ⊢ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46		unass 4093	. . . . . . 7 ⊢ ((𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
47	45, 46	sseqtri 3951	. . . . . 6 ⊢ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
48	44, 47	sstri 3924	. . . . 5 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
49	31, 48	eqsstrdi 3969	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50		coss1 5690	. . . . . . . 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 14367	. . . . . . 7 ⊢ ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
53	51, 52	sstrdi 3927	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54		imass1 5931	. . . . . 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	sstrdi 3927	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
57	49, 56	unssd 4113	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
58	8, 57	unssd 4113	. 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 5896	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61		imaundi 5975	. . . 4 ⊢ (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62		imaundi 5975	. . . . . 6 ⊢ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63		imaco 6071	. . . . . . . 8 ⊢ ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = (𝐹 “ (^◡(t+‘𝐹) “ 𝑈))
64	63	eqcomi 2807	. . . . . . 7 ⊢ (𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈)
65		imaco 6071	. . . . . . . 8 ⊢ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
66	65	eqcomi 2807	. . . . . . 7 ⊢ (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
67	64, 66	uneq12i 4088	. . . . . 6 ⊢ ((𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
68	62, 67	eqtri 2821	. . . . 5 ⊢ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
69	68	uneq2i 4087	. . . 4 ⊢ ((𝐹 “ 𝑈) ∪ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
70	61, 69	eqtri 2821	. . 3 ⊢ (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
71	60, 70	eqtrdi 2849	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
72	58, 71, 59	3sstr4d 3962	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 I cid 5424 ^◡ccnv 5518 ran crn 5520 ↾ cres 5521 “ cima 5522 ∘ ccom 5523 Fun wfun 6318 ‘cfv 6324 t+ctcl 14336

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-seq 13365 df-trcl 14338 df-relexp 14371

This theorem is referenced by: frege133d 40466

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