Theorem frege131d 44208

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 44438. (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 14961	. . . . 5 ⊢ (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3		imass1 6053	. . . . 5 ⊢ (𝐹 ⊆ (t+‘𝐹) → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5		ssun2 4108	. . . . 5 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6		ssun2 4108	. . . . 5 ⊢ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
7	5, 6	sstri 3924	. . . 4 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
8	4, 7	sstrdi 3927	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9		trclfvdecomr 44172	. . . . . . . . . . . 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 5817	. . . . . . . . . 10 ⊢ (𝜑 → ◡(t+‘𝐹) = ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12		cnvun 6093	. . . . . . . . . . 11 ⊢ ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ ◡((t+‘𝐹) ∘ 𝐹))
13		cnvco 5827	. . . . . . . . . . . 12 ⊢ ◡((t+‘𝐹) ∘ 𝐹) = (◡𝐹 ∘ ◡(t+‘𝐹))
14	13	uneq2i 4095	. . . . . . . . . . 11 ⊢ (◡𝐹 ∪ ◡((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))
15	12, 14	eqtri 2762	. . . . . . . . . 10 ⊢ ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))
16	11, 15	eqtrdi 2790	. . . . . . . . 9 ⊢ (𝜑 → ◡(t+‘𝐹) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹))))
17	16	coeq2d 5804	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ ◡(t+‘𝐹)) = (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))))
18		coundi 6198	. . . . . . . . 9 ⊢ (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))) = ((𝐹 ∘ ◡𝐹) ∪ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))))
19		frege131d.fun	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
20		funcocnv2 6792	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
22		coass 6217	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹)) = (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹)))
23	22	eqcomi 2748	. . . . . . . . . . 11 ⊢ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))) = ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹))
24	21	coeq1d 5803	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)))
25	23, 24	eqtrid 2786	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)))
26	21, 25	uneq12d 4099	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ ◡𝐹) ∪ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
27	18, 26	eqtrid 2786	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
28	17, 27	eqtrd 2774	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
29	28	imaeq1d 6011	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))) “ 𝑈))
30		imaundir 6101	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈))
31	29, 30	eqtrdi 2790	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈)))
32		resss 5953	. . . . . . . . 9 ⊢ ( I ↾ ran 𝐹) ⊆ I
33		imass1 6053	. . . . . . . . 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 6026	. . . . . . . 8 ⊢ ( I “ 𝑈) = 𝑈
36	34, 35	sseqtri 3963	. . . . . . 7 ⊢ (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37		imaco 6202	. . . . . . . 8 ⊢ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ (◡(t+‘𝐹) “ 𝑈))
38		imass1 6053	. . . . . . . . . 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 6026	. . . . . . . . 9 ⊢ ( I “ (◡(t+‘𝐹) “ 𝑈)) = (◡(t+‘𝐹) “ 𝑈)
41	39, 40	sseqtri 3963	. . . . . . . 8 ⊢ (( I ↾ ran 𝐹) “ (◡(t+‘𝐹) “ 𝑈)) ⊆ (◡(t+‘𝐹) “ 𝑈)
42	37, 41	eqsstri 3961	. . . . . . 7 ⊢ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈) ⊆ (◡(t+‘𝐹) “ 𝑈)
43		unss12 4117	. . . . . . 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 698	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (◡(t+‘𝐹) “ 𝑈))
45		ssun1 4107	. . . . . . 7 ⊢ (𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46		unass 4101	. . . . . . 7 ⊢ ((𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
47	45, 46	sseqtri 3963	. . . . . 6 ⊢ (𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
48	44, 47	sstri 3924	. . . . 5 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
49	31, 48	eqsstrdi 3959	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50		coss1 5797	. . . . . . . 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 14969	. . . . . . 7 ⊢ ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
53	51, 52	sstrdi 3927	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54		imass1 6053	. . . . . 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 3927	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
57	49, 56	unssd 4121	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
58	8, 57	unssd 4121	. 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 6012	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61		imaundi 6100	. . . 4 ⊢ (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62		imaundi 6100	. . . . . 6 ⊢ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ (◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63		imaco 6202	. . . . . . . 8 ⊢ ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = (𝐹 “ (◡(t+‘𝐹) “ 𝑈))
64	63	eqcomi 2748	. . . . . . 7 ⊢ (𝐹 “ (◡(t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈)
65		imaco 6202	. . . . . . . 8 ⊢ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
66	65	eqcomi 2748	. . . . . . 7 ⊢ (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
67	64, 66	uneq12i 4096	. . . . . 6 ⊢ ((𝐹 “ (◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
68	62, 67	eqtri 2762	. . . . 5 ⊢ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
69	68	uneq2i 4095	. . . 4 ⊢ ((𝐹 “ 𝑈) ∪ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
70	61, 69	eqtri 2762	. . 3 ⊢ (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
71	60, 70	eqtrdi 2790	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
72	58, 71, 59	3sstr4d 3970	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∪ cun 3881   ⊆ wss 3883   I cid 5512  ^◡ccnv 5617  ran crn 5619   ↾ cres 5620   “ cima 5621   ∘ ccom 5622  Fun wfun 6479  ‘cfv 6485  t+ctcl 14938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  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 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-seq 13955  df-trcl 14940  df-relexp 14973

This theorem is referenced by:  frege133d  44209