frege131d - Mathbox for Richard Penner

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Theorem frege131d 44212

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 44442. (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 14964	. . . . 5 ⊢ (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3		imass1 6061	. . . . 5 ⊢ (𝐹 ⊆ (t+‘𝐹) → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5		ssun2 4120	. . . . 5 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6		ssun2 4120	. . . . 5 ⊢ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
7	5, 6	sstri 3932	. . . 4 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
8	4, 7	sstrdi 3935	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9		trclfvdecomr 44176	. . . . . . . . . . . 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 5825	. . . . . . . . . 10 ⊢ (𝜑 → ^◡(t+‘𝐹) = ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12		cnvun 6101	. . . . . . . . . . 11 ⊢ ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ ^◡((t+‘𝐹) ∘ 𝐹))
13		cnvco 5835	. . . . . . . . . . . 12 ⊢ ^◡((t+‘𝐹) ∘ 𝐹) = (^◡𝐹 ∘ ^◡(t+‘𝐹))
14	13	uneq2i 4106	. . . . . . . . . . 11 ⊢ (^◡𝐹 ∪ ^◡((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
15	12, 14	eqtri 2760	. . . . . . . . . 10 ⊢ ^◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
16	11, 15	eqtrdi 2788	. . . . . . . . 9 ⊢ (𝜑 → ^◡(t+‘𝐹) = (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹))))
17	16	coeq2d 5812	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ ^◡(t+‘𝐹)) = (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))))
18		coundi 6206	. . . . . . . . 9 ⊢ (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = ((𝐹 ∘ ^◡𝐹) ∪ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))))
19		frege131d.fun	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
20		funcocnv2 6800	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))
22		coass 6225	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹)) = (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹)))
23	22	eqcomi 2746	. . . . . . . . . . 11 ⊢ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))) = ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹))
24	21	coeq1d 5811	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ∘ ^◡𝐹) ∘ ^◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)))
25	23, 24	eqtrid 2784	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)))
26	21, 25	uneq12d 4110	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ ^◡𝐹) ∪ (𝐹 ∘ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
27	18, 26	eqtrid 2784	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (^◡𝐹 ∪ (^◡𝐹 ∘ ^◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
28	17, 27	eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ^◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))))
29	28	imaeq1d 6019	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))) “ 𝑈))
30		imaundir 6109	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈))
31	29, 30	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)))
32		resss 5961	. . . . . . . . 9 ⊢ ( I ↾ ran 𝐹) ⊆ I
33		imass1 6061	. . . . . . . . 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 6034	. . . . . . . 8 ⊢ ( I “ 𝑈) = 𝑈
36	34, 35	sseqtri 3971	. . . . . . 7 ⊢ (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37		imaco 6210	. . . . . . . 8 ⊢ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ (^◡(t+‘𝐹) “ 𝑈))
38		imass1 6061	. . . . . . . . . 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 6034	. . . . . . . . 9 ⊢ ( I “ (^◡(t+‘𝐹) “ 𝑈)) = (^◡(t+‘𝐹) “ 𝑈)
41	39, 40	sseqtri 3971	. . . . . . . 8 ⊢ (( I ↾ ran 𝐹) “ (^◡(t+‘𝐹) “ 𝑈)) ⊆ (^◡(t+‘𝐹) “ 𝑈)
42	37, 41	eqsstri 3969	. . . . . . 7 ⊢ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈) ⊆ (^◡(t+‘𝐹) “ 𝑈)
43		unss12 4129	. . . . . . 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 693	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈))
45		ssun1 4119	. . . . . . 7 ⊢ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46		unass 4113	. . . . . . 7 ⊢ ((𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
47	45, 46	sseqtri 3971	. . . . . 6 ⊢ (𝑈 ∪ (^◡(t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
48	44, 47	sstri 3932	. . . . 5 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ^◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
49	31, 48	eqsstrdi 3967	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50		coss1 5805	. . . . . . . 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 14972	. . . . . . 7 ⊢ ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
53	51, 52	sstrdi 3935	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54		imass1 6061	. . . . . 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 3935	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
57	49, 56	unssd 4133	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
58	8, 57	unssd 4133	. 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 6020	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61		imaundi 6108	. . . 4 ⊢ (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62		imaundi 6108	. . . . . 6 ⊢ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63		imaco 6210	. . . . . . . 8 ⊢ ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) = (𝐹 “ (^◡(t+‘𝐹) “ 𝑈))
64	63	eqcomi 2746	. . . . . . 7 ⊢ (𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈)
65		imaco 6210	. . . . . . . 8 ⊢ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
66	65	eqcomi 2746	. . . . . . 7 ⊢ (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
67	64, 66	uneq12i 4107	. . . . . 6 ⊢ ((𝐹 “ (^◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
68	62, 67	eqtri 2760	. . . . 5 ⊢ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
69	68	uneq2i 4106	. . . 4 ⊢ ((𝐹 “ 𝑈) ∪ (𝐹 “ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
70	61, 69	eqtri 2760	. . 3 ⊢ (𝐹 “ (𝑈 ∪ ((^◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
71	60, 70	eqtrdi 2788	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ^◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
72	58, 71, 59	3sstr4d 3978	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ∪ cun 3888 ⊆ wss 3890 I cid 5519 ^◡ccnv 5624 ran crn 5626 ↾ cres 5627 “ cima 5628 ∘ ccom 5629 Fun wfun 6487 ‘cfv 6493 t+ctcl 14941

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-n0 12432 df-z 12519 df-uz 12783 df-fz 13456 df-seq 13958 df-trcl 14943 df-relexp 14976

This theorem is referenced by: frege133d 44213

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