frege131d - Mathbox for Richard Penner

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Theorem frege131d 39013

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 39244. (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 14156	. . . . 5 ⊢ (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3		imass1 5754	. . . . 5 ⊢ (𝐹 ⊆ (t+‘𝐹) → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5		ssun2 4000	. . . . 5 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6		ssun2 4000	. . . . 5 ⊢ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
7	5, 6	sstri 3830	. . . 4 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
8	4, 7	syl6ss 3833	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9		trclfvdecomr 38977	. . . . . . . . . . . 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 5543	. . . . . . . . . 10 ⊢ (𝜑 → ^◡(t+‘𝐹) = ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12		cnvun 5792	. . . . . . . . . . 11 ⊢ ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ ^◡((t+‘𝐹) ∘ 𝐹))
13		cnvco 5553	. . . . . . . . . . . 12 ⊢ ^◡((t+‘𝐹) ∘ 𝐹) = (^◡𝐹 ∘ ^◡(t+‘𝐹))
14	13	uneq2i 3987	. . . . . . . . . . 11 ⊢ (^◡𝐹 ∪ ^◡((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
15	12, 14	eqtri 2802	. . . . . . . . . 10 ⊢ ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
16	11, 15	syl6eq 2830	. . . . . . . . 9 ⊢ (𝜑 → ^◡(t+‘𝐹) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹))))
17	16	coeq2d 5530	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ ^◡(t+‘𝐹)) = (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))))
18		coundi 5890	. . . . . . . . 9 ⊢ (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = ((𝐹 ∘ ^◡𝐹) ∪ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))))
19		frege131d.fun	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
20		funcocnv2 6415	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))
22		coass 5908	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹)) = (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
23	22	eqcomi 2787	. . . . . . . . . . 11 ⊢ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))) = ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹))
24	21	coeq1d 5529	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)))
25	23, 24	syl5eq 2826	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)))
26	21, 25	uneq12d 3991	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ ^◡𝐹) ∪ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
27	18, 26	syl5eq 2826	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
28	17, 27	eqtrd 2814	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ^◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
29	28	imaeq1d 5719	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))) “ 𝑈))
30		imaundir 5800	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈))
31	29, 30	syl6eq 2830	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)))
32		resss 5671	. . . . . . . . 9 ⊢ ( I ↾ ran 𝐹) ⊆ I
33		imass1 5754	. . . . . . . . 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 5732	. . . . . . . 8 ⊢ ( I “ 𝑈) = 𝑈
36	34, 35	sseqtri 3856	. . . . . . 7 ⊢ (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37		imaco 5894	. . . . . . . 8 ⊢ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ (^◡(t+‘𝐹) “ 𝑈))
38		imass1 5754	. . . . . . . . . 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 5732	. . . . . . . . 9 ⊢ ( I “ (^◡(t+‘𝐹) “ 𝑈)) = (^◡(t+‘𝐹) “ 𝑈)
41	39, 40	sseqtri 3856	. . . . . . . 8 ⊢ (( I ↾ ran 𝐹) “ (^◡(t+‘𝐹) “ 𝑈)) ⊆ (^◡(t+‘𝐹) “ 𝑈)
42	37, 41	eqsstri 3854	. . . . . . 7 ⊢ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈) ⊆ (^◡(t+‘𝐹) “ 𝑈)
43		unss12 4008	. . . . . . 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 682	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈))
45		ssun1 3999	. . . . . . 7 ⊢ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46		unass 3993	. . . . . . 7 ⊢ ((𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
47	45, 46	sseqtri 3856	. . . . . 6 ⊢ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
48	44, 47	sstri 3830	. . . . 5 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
49	31, 48	syl6eqss 3874	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50		coss1 5523	. . . . . . . 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 14164	. . . . . . 7 ⊢ ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
53	51, 52	syl6ss 3833	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54		imass1 5754	. . . . . 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 3833	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
57	49, 56	unssd 4012	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
58	8, 57	unssd 4012	. 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 5720	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61		imaundi 5799	. . . 4 ⊢ (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62		imaundi 5799	. . . . . 6 ⊢ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63		imaco 5894	. . . . . . . 8 ⊢ ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = (𝐹 “ (^◡(t+‘𝐹) “ 𝑈))
64	63	eqcomi 2787	. . . . . . 7 ⊢ (𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈)
65		imaco 5894	. . . . . . . 8 ⊢ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
66	65	eqcomi 2787	. . . . . . 7 ⊢ (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
67	64, 66	uneq12i 3988	. . . . . 6 ⊢ ((𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
68	62, 67	eqtri 2802	. . . . 5 ⊢ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
69	68	uneq2i 3987	. . . 4 ⊢ ((𝐹 “ 𝑈) ∪ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
70	61, 69	eqtri 2802	. . 3 ⊢ (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
71	60, 70	syl6eq 2830	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
72	58, 71, 59	3sstr4d 3867	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∪ cun 3790 ⊆ wss 3792 I cid 5260 ^◡ccnv 5354 ran crn 5356 ↾ cres 5357 “ cima 5358 ∘ ccom 5359 Fun wfun 6129 ‘cfv 6135 t+ctcl 14133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-seq 13120 df-trcl 14135 df-relexp 14168

This theorem is referenced by: frege133d 39014

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