Theorem frege131d 43947

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 44177. (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 14929	. . . . 5 ⊢ (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3		imass1 6058	. . . . 5 ⊢ (𝐹 ⊆ (t+‘𝐹) → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5		ssun2 4129	. . . . 5 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6		ssun2 4129	. . . . 5 ⊢ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
7	5, 6	sstri 3941	. . . 4 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
8	4, 7	sstrdi 3944	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9		trclfvdecomr 43911	. . . . . . . . . . . 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 5822	. . . . . . . . . 10 ⊢ (𝜑 → ◡(t+‘𝐹) = ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12		cnvun 6098	. . . . . . . . . . 11 ⊢ ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ ◡((t+‘𝐹) ∘ 𝐹))
13		cnvco 5832	. . . . . . . . . . . 12 ⊢ ◡((t+‘𝐹) ∘ 𝐹) = (◡𝐹 ∘ ◡(t+‘𝐹))
14	13	uneq2i 4115	. . . . . . . . . . 11 ⊢ (◡𝐹 ∪ ◡((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))
15	12, 14	eqtri 2757	. . . . . . . . . 10 ⊢ ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))
16	11, 15	eqtrdi 2785	. . . . . . . . 9 ⊢ (𝜑 → ◡(t+‘𝐹) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹))))
17	16	coeq2d 5809	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ ◡(t+‘𝐹)) = (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))))
18		coundi 6203	. . . . . . . . 9 ⊢ (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))) = ((𝐹 ∘ ◡𝐹) ∪ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))))
19		frege131d.fun	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
20		funcocnv2 6797	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
22		coass 6222	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹)) = (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹)))
23	22	eqcomi 2743	. . . . . . . . . . 11 ⊢ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))) = ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹))
24	21	coeq1d 5808	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)))
25	23, 24	eqtrid 2781	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)))
26	21, 25	uneq12d 4119	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ ◡𝐹) ∪ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
27	18, 26	eqtrid 2781	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
28	17, 27	eqtrd 2769	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
29	28	imaeq1d 6016	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))) “ 𝑈))
30		imaundir 6106	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈))
31	29, 30	eqtrdi 2785	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈)))
32		resss 5958	. . . . . . . . 9 ⊢ ( I ↾ ran 𝐹) ⊆ I
33		imass1 6058	. . . . . . . . 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 6031	. . . . . . . 8 ⊢ ( I “ 𝑈) = 𝑈
36	34, 35	sseqtri 3980	. . . . . . 7 ⊢ (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37		imaco 6207	. . . . . . . 8 ⊢ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ (◡(t+‘𝐹) “ 𝑈))
38		imass1 6058	. . . . . . . . . 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 6031	. . . . . . . . 9 ⊢ ( I “ (◡(t+‘𝐹) “ 𝑈)) = (◡(t+‘𝐹) “ 𝑈)
41	39, 40	sseqtri 3980	. . . . . . . 8 ⊢ (( I ↾ ran 𝐹) “ (◡(t+‘𝐹) “ 𝑈)) ⊆ (◡(t+‘𝐹) “ 𝑈)
42	37, 41	eqsstri 3978	. . . . . . 7 ⊢ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈) ⊆ (◡(t+‘𝐹) “ 𝑈)
43		unss12 4138	. . . . . . 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 4128	. . . . . . 7 ⊢ (𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46		unass 4122	. . . . . . 7 ⊢ ((𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
47	45, 46	sseqtri 3980	. . . . . 6 ⊢ (𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
48	44, 47	sstri 3941	. . . . 5 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
49	31, 48	eqsstrdi 3976	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50		coss1 5802	. . . . . . . 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 14937	. . . . . . 7 ⊢ ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
53	51, 52	sstrdi 3944	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54		imass1 6058	. . . . . 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 3944	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
57	49, 56	unssd 4142	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
58	8, 57	unssd 4142	. 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 6017	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61		imaundi 6105	. . . 4 ⊢ (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62		imaundi 6105	. . . . . 6 ⊢ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ (◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63		imaco 6207	. . . . . . . 8 ⊢ ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = (𝐹 “ (◡(t+‘𝐹) “ 𝑈))
64	63	eqcomi 2743	. . . . . . 7 ⊢ (𝐹 “ (◡(t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈)
65		imaco 6207	. . . . . . . 8 ⊢ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
66	65	eqcomi 2743	. . . . . . 7 ⊢ (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
67	64, 66	uneq12i 4116	. . . . . 6 ⊢ ((𝐹 “ (◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
68	62, 67	eqtri 2757	. . . . 5 ⊢ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
69	68	uneq2i 4115	. . . 4 ⊢ ((𝐹 “ 𝑈) ∪ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
70	61, 69	eqtri 2757	. . 3 ⊢ (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
71	60, 70	eqtrdi 2785	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
72	58, 71, 59	3sstr4d 3987	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∪ cun 3897   ⊆ wss 3899   I cid 5516  ^◡ccnv 5621  ran crn 5623   ↾ cres 5624   “ cima 5625   ∘ ccom 5626  Fun wfun 6484  ‘cfv 6490  t+ctcl 14906

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-n0 12400  df-z 12487  df-uz 12750  df-fz 13422  df-seq 13923  df-trcl 14908  df-relexp 14941

This theorem is referenced by:  frege133d  43948