Theorem restutop 24123

Description: Restriction of a topology induced by an uniform structure. (Contributed by Thierry Arnoux, 12-Dec-2017.)

Assertion

Ref	Expression
restutop	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))

Proof of Theorem restutop

Dummy variables 𝑎 𝑏 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋))
2		fvexd 6837	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (unifTop‘𝑈) ∈ V)
3		elfvex 6858	. . . . . . . . 9 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ V)
5		simpr 484	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
6	4, 5	ssexd 5263	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
7		elrest 17331	. . . . . . 7 ⊢ (((unifTop‘𝑈) ∈ V ∧ 𝐴 ∈ V) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)))
8	2, 6, 7	syl2anc 584	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)))
9	8	biimpa 476	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴))
10		inss2 4189	. . . . . . 7 ⊢ (𝑎 ∩ 𝐴) ⊆ 𝐴
11		sseq1 3961	. . . . . . 7 ⊢ (𝑏 = (𝑎 ∩ 𝐴) → (𝑏 ⊆ 𝐴 ↔ (𝑎 ∩ 𝐴) ⊆ 𝐴))
12	10, 11	mpbiri 258	. . . . . 6 ⊢ (𝑏 = (𝑎 ∩ 𝐴) → 𝑏 ⊆ 𝐴)
13	12	rexlimivw 3126	. . . . 5 ⊢ (∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴) → 𝑏 ⊆ 𝐴)
14	9, 13	syl 17	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏 ⊆ 𝐴)
15		simp-5l 784	. . . . . . . . . 10 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑈 ∈ (UnifOn‘𝑋))
16	15	ad2antrr 726	. . . . . . . . 9 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
17	6	ad6antr 736	. . . . . . . . . 10 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝐴 ∈ V)
18	17, 17	xpexd 7687	. . . . . . . . 9 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → (𝐴 × 𝐴) ∈ V)
19		simplr 768	. . . . . . . . 9 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑢 ∈ 𝑈)
20		elrestr 17332	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑢 ∈ 𝑈) → (𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
21	16, 18, 19, 20	syl3anc 1373	. . . . . . . 8 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → (𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
22		inss1 4188	. . . . . . . . . . . . 13 ⊢ (𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢
23		imass1 6052	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}))
24	22, 23	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥})
25		sstr 3944	. . . . . . . . . . . 12 ⊢ ((((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎)
26	24, 25	mpan 690	. . . . . . . . . . 11 ⊢ ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎)
27		imassrn 6022	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝑢 ∩ (𝐴 × 𝐴))
28		rnin 6095	. . . . . . . . . . . . . . 15 ⊢ ran (𝑢 ∩ (𝐴 × 𝐴)) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴))
29	27, 28	sstri 3945	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴))
30		inss2 4189	. . . . . . . . . . . . . 14 ⊢ (ran 𝑢 ∩ ran (𝐴 × 𝐴)) ⊆ ran (𝐴 × 𝐴)
31	29, 30	sstri 3945	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝐴 × 𝐴)
32		rnxpid 6122	. . . . . . . . . . . . 13 ⊢ ran (𝐴 × 𝐴) = 𝐴
33	31, 32	sseqtri 3984	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴
34	33	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴)
35	26, 34	ssind 4192	. . . . . . . . . 10 ⊢ ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑎 ∩ 𝐴))
36	35	adantl 481	. . . . . . . . 9 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑎 ∩ 𝐴))
37		simpllr 775	. . . . . . . . 9 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑏 = (𝑎 ∩ 𝐴))
38	36, 37	sseqtrrd 3973	. . . . . . . 8 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏)
39		imaeq1 6006	. . . . . . . . . 10 ⊢ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → (𝑣 “ {𝑥}) = ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}))
40	39	sseq1d 3967	. . . . . . . . 9 ⊢ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏))
41	40	rspcev 3577	. . . . . . . 8 ⊢ (((𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏) → ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
42	21, 38, 41	syl2anc 584	. . . . . . 7 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
43		simplr 768	. . . . . . . 8 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑎 ∈ (unifTop‘𝑈))
44		simpllr 775	. . . . . . . . . 10 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑥 ∈ 𝑏)
45		simpr 484	. . . . . . . . . 10 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑏 = (𝑎 ∩ 𝐴))
46	44, 45	eleqtrd 2830	. . . . . . . . 9 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑥 ∈ (𝑎 ∩ 𝐴))
47	46	elin1d 4155	. . . . . . . 8 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑥 ∈ 𝑎)
48		elutop 24119	. . . . . . . . . 10 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑎 ∈ (unifTop‘𝑈) ↔ (𝑎 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)))
49	48	simplbda 499	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) → ∀𝑥 ∈ 𝑎 ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
50	49	r19.21bi 3221	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑥 ∈ 𝑎) → ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
51	15, 43, 47, 50	syl21anc 837	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎)
52	42, 51	r19.29a 3137	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
53	9	adantr 480	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴))
54	52, 53	r19.29a 3137	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) → ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
55	54	ralrimiva 3121	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)
56		trust 24115	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
57		elutop 24119	. . . . . 6 ⊢ ((𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))) ↔ (𝑏 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)))
58	56, 57	syl 17	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))) ↔ (𝑏 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)))
59	58	biimpar 477	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)) → 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))
60	1, 14, 55, 59	syl12anc 836	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))
61	60	ex 412	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) → 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))))
62	61	ssrdv 3941	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  {csn 4577   × cxp 5617  ran crn 5620   “ cima 5622  ‘cfv 6482  (class class class)co 7349   ↾_t crest 17324  UnifOncust 24085  unifTopcutop 24116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-rest 17326  df-ust 24086  df-utop 24117

This theorem is referenced by:  restutopopn  24124