frege131d - Mathbox for Richard Penner

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Theorem frege131d 43867

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 44097. (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 14915	. . . . 5 ⊢ (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3		imass1 6049	. . . . 5 ⊢ (𝐹 ⊆ (t+‘𝐹) → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5		ssun2 4126	. . . . 5 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6		ssun2 4126	. . . . 5 ⊢ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
7	5, 6	sstri 3939	. . . 4 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
8	4, 7	sstrdi 3942	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9		trclfvdecomr 43831	. . . . . . . . . . . 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 5814	. . . . . . . . . 10 ⊢ (𝜑 → ^◡(t+‘𝐹) = ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12		cnvun 6089	. . . . . . . . . . 11 ⊢ ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ ^◡((t+‘𝐹) ∘ 𝐹))
13		cnvco 5824	. . . . . . . . . . . 12 ⊢ ^◡((t+‘𝐹) ∘ 𝐹) = (^◡𝐹 ∘ ^◡(t+‘𝐹))
14	13	uneq2i 4112	. . . . . . . . . . 11 ⊢ (^◡𝐹 ∪ ^◡((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
15	12, 14	eqtri 2754	. . . . . . . . . 10 ⊢ ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
16	11, 15	eqtrdi 2782	. . . . . . . . 9 ⊢ (𝜑 → ^◡(t+‘𝐹) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹))))
17	16	coeq2d 5801	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ ^◡(t+‘𝐹)) = (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))))
18		coundi 6194	. . . . . . . . 9 ⊢ (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = ((𝐹 ∘ ^◡𝐹) ∪ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))))
19		frege131d.fun	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
20		funcocnv2 6788	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))
22		coass 6213	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹)) = (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
23	22	eqcomi 2740	. . . . . . . . . . 11 ⊢ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))) = ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹))
24	21	coeq1d 5800	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)))
25	23, 24	eqtrid 2778	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)))
26	21, 25	uneq12d 4116	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ ^◡𝐹) ∪ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
27	18, 26	eqtrid 2778	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
28	17, 27	eqtrd 2766	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ^◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
29	28	imaeq1d 6007	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))) “ 𝑈))
30		imaundir 6097	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈))
31	29, 30	eqtrdi 2782	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)))
32		resss 5949	. . . . . . . . 9 ⊢ ( I ↾ ran 𝐹) ⊆ I
33		imass1 6049	. . . . . . . . 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 6022	. . . . . . . 8 ⊢ ( I “ 𝑈) = 𝑈
36	34, 35	sseqtri 3978	. . . . . . 7 ⊢ (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37		imaco 6198	. . . . . . . 8 ⊢ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ (^◡(t+‘𝐹) “ 𝑈))
38		imass1 6049	. . . . . . . . . 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 6022	. . . . . . . . 9 ⊢ ( I “ (^◡(t+‘𝐹) “ 𝑈)) = (^◡(t+‘𝐹) “ 𝑈)
41	39, 40	sseqtri 3978	. . . . . . . 8 ⊢ (( I ↾ ran 𝐹) “ (^◡(t+‘𝐹) “ 𝑈)) ⊆ (^◡(t+‘𝐹) “ 𝑈)
42	37, 41	eqsstri 3976	. . . . . . 7 ⊢ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈) ⊆ (^◡(t+‘𝐹) “ 𝑈)
43		unss12 4135	. . . . . . 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 692	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈))
45		ssun1 4125	. . . . . . 7 ⊢ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46		unass 4119	. . . . . . 7 ⊢ ((𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
47	45, 46	sseqtri 3978	. . . . . 6 ⊢ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
48	44, 47	sstri 3939	. . . . 5 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
49	31, 48	eqsstrdi 3974	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50		coss1 5794	. . . . . . . 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 14923	. . . . . . 7 ⊢ ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
53	51, 52	sstrdi 3942	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54		imass1 6049	. . . . . 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 3942	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
57	49, 56	unssd 4139	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
58	8, 57	unssd 4139	. 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 6008	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61		imaundi 6096	. . . 4 ⊢ (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62		imaundi 6096	. . . . . 6 ⊢ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63		imaco 6198	. . . . . . . 8 ⊢ ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = (𝐹 “ (^◡(t+‘𝐹) “ 𝑈))
64	63	eqcomi 2740	. . . . . . 7 ⊢ (𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈)
65		imaco 6198	. . . . . . . 8 ⊢ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
66	65	eqcomi 2740	. . . . . . 7 ⊢ (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
67	64, 66	uneq12i 4113	. . . . . 6 ⊢ ((𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
68	62, 67	eqtri 2754	. . . . 5 ⊢ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
69	68	uneq2i 4112	. . . 4 ⊢ ((𝐹 “ 𝑈) ∪ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
70	61, 69	eqtri 2754	. . 3 ⊢ (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
71	60, 70	eqtrdi 2782	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
72	58, 71, 59	3sstr4d 3985	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 Vcvv 3436 ∪ cun 3895 ⊆ wss 3897 I cid 5508 ^◡ccnv 5613 ran crn 5615 ↾ cres 5616 “ cima 5617 ∘ ccom 5618 Fun wfun 6475 ‘cfv 6481 t+ctcl 14892

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-seq 13909 df-trcl 14894 df-relexp 14927

This theorem is referenced by: frege133d 43868

W3C validator