Theorem frege131d 41372

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 41602. (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 14719	. . . . 5 ⊢ (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3		imass1 6009	. . . . 5 ⊢ (𝐹 ⊆ (t+‘𝐹) → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5		ssun2 4107	. . . . 5 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6		ssun2 4107	. . . . 5 ⊢ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
7	5, 6	sstri 3930	. . . 4 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
8	4, 7	sstrdi 3933	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9		trclfvdecomr 41336	. . . . . . . . . . . 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 5784	. . . . . . . . . 10 ⊢ (𝜑 → ◡(t+‘𝐹) = ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12		cnvun 6046	. . . . . . . . . . 11 ⊢ ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ ◡((t+‘𝐹) ∘ 𝐹))
13		cnvco 5794	. . . . . . . . . . . 12 ⊢ ◡((t+‘𝐹) ∘ 𝐹) = (◡𝐹 ∘ ◡(t+‘𝐹))
14	13	uneq2i 4094	. . . . . . . . . . 11 ⊢ (◡𝐹 ∪ ◡((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))
15	12, 14	eqtri 2766	. . . . . . . . . 10 ⊢ ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))
16	11, 15	eqtrdi 2794	. . . . . . . . 9 ⊢ (𝜑 → ◡(t+‘𝐹) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹))))
17	16	coeq2d 5771	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ ◡(t+‘𝐹)) = (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))))
18		coundi 6151	. . . . . . . . 9 ⊢ (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))) = ((𝐹 ∘ ◡𝐹) ∪ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))))
19		frege131d.fun	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
20		funcocnv2 6741	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
22		coass 6169	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹)) = (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹)))
23	22	eqcomi 2747	. . . . . . . . . . 11 ⊢ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))) = ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹))
24	21	coeq1d 5770	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)))
25	23, 24	eqtrid 2790	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)))
26	21, 25	uneq12d 4098	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ ◡𝐹) ∪ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
27	18, 26	eqtrid 2790	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
28	17, 27	eqtrd 2778	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
29	28	imaeq1d 5968	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))) “ 𝑈))
30		imaundir 6054	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈))
31	29, 30	eqtrdi 2794	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈)))
32		resss 5916	. . . . . . . . 9 ⊢ ( I ↾ ran 𝐹) ⊆ I
33		imass1 6009	. . . . . . . . 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 5982	. . . . . . . 8 ⊢ ( I “ 𝑈) = 𝑈
36	34, 35	sseqtri 3957	. . . . . . 7 ⊢ (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37		imaco 6155	. . . . . . . 8 ⊢ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ (◡(t+‘𝐹) “ 𝑈))
38		imass1 6009	. . . . . . . . . 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 5982	. . . . . . . . 9 ⊢ ( I “ (◡(t+‘𝐹) “ 𝑈)) = (◡(t+‘𝐹) “ 𝑈)
41	39, 40	sseqtri 3957	. . . . . . . 8 ⊢ (( I ↾ ran 𝐹) “ (◡(t+‘𝐹) “ 𝑈)) ⊆ (◡(t+‘𝐹) “ 𝑈)
42	37, 41	eqsstri 3955	. . . . . . 7 ⊢ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈) ⊆ (◡(t+‘𝐹) “ 𝑈)
43		unss12 4116	. . . . . . 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 689	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (◡(t+‘𝐹) “ 𝑈))
45		ssun1 4106	. . . . . . 7 ⊢ (𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46		unass 4100	. . . . . . 7 ⊢ ((𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
47	45, 46	sseqtri 3957	. . . . . 6 ⊢ (𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
48	44, 47	sstri 3930	. . . . 5 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
49	31, 48	eqsstrdi 3975	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50		coss1 5764	. . . . . . . 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 14727	. . . . . . 7 ⊢ ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
53	51, 52	sstrdi 3933	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54		imass1 6009	. . . . . 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 3933	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
57	49, 56	unssd 4120	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
58	8, 57	unssd 4120	. 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 5969	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61		imaundi 6053	. . . 4 ⊢ (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62		imaundi 6053	. . . . . 6 ⊢ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ (◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63		imaco 6155	. . . . . . . 8 ⊢ ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = (𝐹 “ (◡(t+‘𝐹) “ 𝑈))
64	63	eqcomi 2747	. . . . . . 7 ⊢ (𝐹 “ (◡(t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈)
65		imaco 6155	. . . . . . . 8 ⊢ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
66	65	eqcomi 2747	. . . . . . 7 ⊢ (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
67	64, 66	uneq12i 4095	. . . . . 6 ⊢ ((𝐹 “ (◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
68	62, 67	eqtri 2766	. . . . 5 ⊢ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
69	68	uneq2i 4094	. . . 4 ⊢ ((𝐹 “ 𝑈) ∪ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
70	61, 69	eqtri 2766	. . 3 ⊢ (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
71	60, 70	eqtrdi 2794	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
72	58, 71, 59	3sstr4d 3968	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887   I cid 5488  ^◡ccnv 5588  ran crn 5590   ↾ cres 5591   “ cima 5592   ∘ ccom 5593  Fun wfun 6427  ‘cfv 6433  t+ctcl 14696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-seq 13722  df-trcl 14698  df-relexp 14731

This theorem is referenced by:  frege133d  41373