Theorem frege131d 44904

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 45134. (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 15173	. . . . 5 ⊢ (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3		imass1 6059	. . . . 5 ⊢ (𝐹 ⊆ (t+‘𝐹) → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
4	1, 2, 3	3syl 19	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5		ssun2 4179	. . . . 5 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6		ssun2 4179	. . . . 5 ⊢ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
7	5, 6	sstri 3921	. . . 4 ⊢ ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
8	4, 7	sstrdi 3924	. . 3 ⊢ (𝜑 → (𝐹 “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9		trclfvdecomr 44868	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ V → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
10	1, 9	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
11	10	cnveqd 5857	. . . . . . . . . 10 ⊢ (𝜑 → ◡(t+‘𝐹) = ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12		cnvun 6138	. . . . . . . . . . 11 ⊢ ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ ◡((t+‘𝐹) ∘ 𝐹))
13		cnvco 5871	. . . . . . . . . . . 12 ⊢ ◡((t+‘𝐹) ∘ 𝐹) = (◡𝐹 ∘ ◡(t+‘𝐹))
14	13	uneq2i 4166	. . . . . . . . . . 11 ⊢ (◡𝐹 ∪ ◡((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))
15	12, 14	eqtri 2786	. . . . . . . . . 10 ⊢ ◡(𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))
16	11, 15	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝜑 → ◡(t+‘𝐹) = (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹))))
17	16	coeq2d 5844	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ ◡(t+‘𝐹)) = (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))))
18		coundi 6254	. . . . . . . . 9 ⊢ (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))) = ((𝐹 ∘ ◡𝐹) ∪ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))))
19		frege131d.fun	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
20		funcocnv2 6858	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
21	19, 20	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
22		coass 6273	. . . . . . . . . . . 12 ⊢ ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹)) = (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹)))
23	22	eqcomi 2772	. . . . . . . . . . 11 ⊢ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))) = ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹))
24	21	coeq1d 5843	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ∘ ◡𝐹) ∘ ◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)))
25	23, 24	eqtrid 2810	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)))
26	21, 25	uneq12d 4170	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ ◡𝐹) ∪ (𝐹 ∘ (◡𝐹 ∘ ◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
27	18, 26	eqtrid 2810	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (◡𝐹 ∪ (◡𝐹 ∘ ◡(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
28	17, 27	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ◡(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))))
29	28	imaeq1d 6056	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))) “ 𝑈))
30		imaundir 6147	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈))
31	29, 30	eqtrdi 2814	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈)))
32		resss 5996	. . . . . . . . 9 ⊢ ( I ↾ ran 𝐹) ⊆ I
33		imass1 6059	. . . . . . . . 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 6077	. . . . . . . 8 ⊢ ( I “ 𝑈) = 𝑈
36	34, 35	sseqtri 3960	. . . . . . 7 ⊢ (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37		imaco 6258	. . . . . . . 8 ⊢ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ (◡(t+‘𝐹) “ 𝑈))
38		imass1 6059	. . . . . . . . . 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 6077	. . . . . . . . 9 ⊢ ( I “ (◡(t+‘𝐹) “ 𝑈)) = (◡(t+‘𝐹) “ 𝑈)
41	39, 40	sseqtri 3960	. . . . . . . 8 ⊢ (( I ↾ ran 𝐹) “ (◡(t+‘𝐹) “ 𝑈)) ⊆ (◡(t+‘𝐹) “ 𝑈)
42	37, 41	eqsstri 3958	. . . . . . 7 ⊢ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈) ⊆ (◡(t+‘𝐹) “ 𝑈)
43		unss12 4188	. . . . . . 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 705	. . . . . 6 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (◡(t+‘𝐹) “ 𝑈))
45		ssun1 4178	. . . . . . 7 ⊢ (𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46		unass 4172	. . . . . . 7 ⊢ ((𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
47	45, 46	sseqtri 3960	. . . . . 6 ⊢ (𝑈 ∪ (◡(t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
48	44, 47	sstri 3921	. . . . 5 ⊢ ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ ◡(t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
49	31, 48	eqsstrdi 3956	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50		coss1 5837	. . . . . . . 8 ⊢ (𝐹 ⊆ (t+‘𝐹) → (𝐹 ∘ (t+‘𝐹)) ⊆ ((t+‘𝐹) ∘ (t+‘𝐹)))
51	1, 2, 50	3syl 19	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ ((t+‘𝐹) ∘ (t+‘𝐹)))
52		trclfvcotrg 15181	. . . . . . 7 ⊢ ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
53	51, 52	sstrdi 3924	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54		imass1 6059	. . . . . 6 ⊢ ((𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹) → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
55	53, 54	syl 18	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
56	55, 7	sstrdi 3924	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
57	49, 56	unssd 4192	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
58	8, 57	unssd 4192	. 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 6057	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61		imaundi 6146	. . . 4 ⊢ (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62		imaundi 6146	. . . . . 6 ⊢ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ (◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63		imaco 6258	. . . . . . . 8 ⊢ ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) = (𝐹 “ (◡(t+‘𝐹) “ 𝑈))
64	63	eqcomi 2772	. . . . . . 7 ⊢ (𝐹 “ (◡(t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈)
65		imaco 6258	. . . . . . . 8 ⊢ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
66	65	eqcomi 2772	. . . . . . 7 ⊢ (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
67	64, 66	uneq12i 4167	. . . . . 6 ⊢ ((𝐹 “ (◡(t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
68	62, 67	eqtri 2786	. . . . 5 ⊢ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
69	68	uneq2i 4166	. . . 4 ⊢ ((𝐹 “ 𝑈) ∪ (𝐹 “ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
70	61, 69	eqtri 2786	. . 3 ⊢ (𝐹 “ (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
71	60, 70	eqtrdi 2814	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) = ((𝐹 “ 𝑈) ∪ (((𝐹 ∘ ◡(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
72	58, 71, 59	3sstr4d 3967	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  Vcvv 3453   ⊆ wss 3899   ∪ cun 4151   I cid 5548  ^◡ccnv 5653  ran crn 5655   ↾ cres 5656   “ cima 5657   ∘ ccom 5658  Fun wfun 6540  ‘cfv 6546  t+ctcl 15150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-er 8728  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-nn 12346  df-2 12415  df-n0 12617  df-z 12704  df-uz 12976  df-fz 13650  df-seq 14155  df-trcl 15152  df-relexp 15185

This theorem is used by:  frege133d  44905