Theorem cnpconn 32698

Description: An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)

Hypothesis

Ref	Expression
cnpconn.2	⊢ 𝑌 = ∪ 𝐾

Assertion

Ref	Expression
cnpconn	⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)

Proof of Theorem cnpconn

Dummy variables 𝑓 𝑔 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntop2 21931	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2	1	3ad2ant3 1133	. 2 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3		eqid 2759	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	pconncn 32692	. . . . . . . 8 ⊢ ((𝐽 ∈ PConn ∧ 𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
5	4	3expb 1118	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
6	5	3ad2antl1 1183	. . . . . 6 ⊢ (((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
7		simprl 771	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → 𝑔 ∈ (II Cn 𝐽))
8		simpll3 1212	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → 𝐹 ∈ (𝐽 Cn 𝐾))
9		cnco 21956	. . . . . . . 8 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ 𝑔) ∈ (II Cn 𝐾))
10	7, 8, 9	syl2anc 588	. . . . . . 7 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝐹 ∘ 𝑔) ∈ (II Cn 𝐾))
11		iiuni 23572	. . . . . . . . . . 11 ⊢ (0[,]1) = ∪ II
12	11, 3	cnf 21936	. . . . . . . . . 10 ⊢ (𝑔 ∈ (II Cn 𝐽) → 𝑔:(0[,]1)⟶∪ 𝐽)
13	7, 12	syl 17	. . . . . . . . 9 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → 𝑔:(0[,]1)⟶∪ 𝐽)
14		0elunit 12891	. . . . . . . . 9 ⊢ 0 ∈ (0[,]1)
15		fvco3 6749	. . . . . . . . 9 ⊢ ((𝑔:(0[,]1)⟶∪ 𝐽 ∧ 0 ∈ (0[,]1)) → ((𝐹 ∘ 𝑔)‘0) = (𝐹‘(𝑔‘0)))
16	13, 14, 15	sylancl 590	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → ((𝐹 ∘ 𝑔)‘0) = (𝐹‘(𝑔‘0)))
17		simprrl 781	. . . . . . . . 9 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝑔‘0) = 𝑢)
18	17	fveq2d 6660	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝐹‘(𝑔‘0)) = (𝐹‘𝑢))
19	16, 18	eqtrd 2794	. . . . . . 7 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → ((𝐹 ∘ 𝑔)‘0) = (𝐹‘𝑢))
20		1elunit 12892	. . . . . . . . 9 ⊢ 1 ∈ (0[,]1)
21		fvco3 6749	. . . . . . . . 9 ⊢ ((𝑔:(0[,]1)⟶∪ 𝐽 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝑔)‘1) = (𝐹‘(𝑔‘1)))
22	13, 20, 21	sylancl 590	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → ((𝐹 ∘ 𝑔)‘1) = (𝐹‘(𝑔‘1)))
23		simprrr 782	. . . . . . . . 9 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝑔‘1) = 𝑣)
24	23	fveq2d 6660	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝐹‘(𝑔‘1)) = (𝐹‘𝑣))
25	22, 24	eqtrd 2794	. . . . . . 7 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → ((𝐹 ∘ 𝑔)‘1) = (𝐹‘𝑣))
26		fveq1 6655	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘0) = ((𝐹 ∘ 𝑔)‘0))
27	26	eqeq1d 2761	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘0) = (𝐹‘𝑢) ↔ ((𝐹 ∘ 𝑔)‘0) = (𝐹‘𝑢)))
28		fveq1 6655	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘1) = ((𝐹 ∘ 𝑔)‘1))
29	28	eqeq1d 2761	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘1) = (𝐹‘𝑣) ↔ ((𝐹 ∘ 𝑔)‘1) = (𝐹‘𝑣)))
30	27, 29	anbi12d 634	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)) ↔ (((𝐹 ∘ 𝑔)‘0) = (𝐹‘𝑢) ∧ ((𝐹 ∘ 𝑔)‘1) = (𝐹‘𝑣))))
31	30	rspcev 3542	. . . . . . 7 ⊢ (((𝐹 ∘ 𝑔) ∈ (II Cn 𝐾) ∧ (((𝐹 ∘ 𝑔)‘0) = (𝐹‘𝑢) ∧ ((𝐹 ∘ 𝑔)‘1) = (𝐹‘𝑣))) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)))
32	10, 19, 25, 31	syl12anc 836	. . . . . 6 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)))
33	6, 32	rexlimddv 3216	. . . . 5 ⊢ (((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)))
34	33	ralrimivva 3121	. . . 4 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 ∈ ∪ 𝐽∀𝑣 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)))
35		cnpconn.2	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐾
36	3, 35	cnf 21936	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
37	36	3ad2ant3 1133	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌)
38		forn 6577	. . . . . . . 8 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
39	38	3ad2ant2 1132	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = 𝑌)
40		dffo2 6578	. . . . . . 7 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹:∪ 𝐽⟶𝑌 ∧ ran 𝐹 = 𝑌))
41	37, 39, 40	sylanbrc 587	. . . . . 6 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽–onto→𝑌)
42		eqeq2 2771	. . . . . . . . 9 ⊢ ((𝐹‘𝑣) = 𝑦 → ((𝑓‘1) = (𝐹‘𝑣) ↔ (𝑓‘1) = 𝑦))
43	42	anbi2d 632	. . . . . . . 8 ⊢ ((𝐹‘𝑣) = 𝑦 → (((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)) ↔ ((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦)))
44	43	rexbidv 3222	. . . . . . 7 ⊢ ((𝐹‘𝑣) = 𝑦 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦)))
45	44	cbvfo 7035	. . . . . 6 ⊢ (𝐹:∪ 𝐽–onto→𝑌 → (∀𝑣 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)) ↔ ∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦)))
46	41, 45	syl 17	. . . . 5 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (∀𝑣 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)) ↔ ∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦)))
47	46	ralbidv 3127	. . . 4 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (∀𝑢 ∈ ∪ 𝐽∀𝑣 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)) ↔ ∀𝑢 ∈ ∪ 𝐽∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦)))
48	34, 47	mpbid 235	. . 3 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 ∈ ∪ 𝐽∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦))
49		eqeq2 2771	. . . . . . . 8 ⊢ ((𝐹‘𝑢) = 𝑥 → ((𝑓‘0) = (𝐹‘𝑢) ↔ (𝑓‘0) = 𝑥))
50	49	anbi1d 633	. . . . . . 7 ⊢ ((𝐹‘𝑢) = 𝑥 → (((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
51	50	rexbidv 3222	. . . . . 6 ⊢ ((𝐹‘𝑢) = 𝑥 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
52	51	ralbidv 3127	. . . . 5 ⊢ ((𝐹‘𝑢) = 𝑥 → (∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
53	52	cbvfo 7035	. . . 4 ⊢ (𝐹:∪ 𝐽–onto→𝑌 → (∀𝑢 ∈ ∪ 𝐽∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
54	41, 53	syl 17	. . 3 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (∀𝑢 ∈ ∪ 𝐽∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
55	48, 54	mpbid 235	. 2 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
56	35	ispconn 32691	. 2 ⊢ (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
57	2, 55, 56	sylanbrc 587	1 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  ∪ cuni 4796  ran crn 5523   ∘ ccom 5526  ⟶wf 6329  –onto→wfo 6331  ‘cfv 6333  (class class class)co 7148  0cc0 10565  1c1 10566  [,]cicc 12772  Topctop 21583   Cn ccn 21914  IIcii 23566  PConncpconn 32687

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457  ax-cnex 10621  ax-resscn 10622  ax-1cn 10623  ax-icn 10624  ax-addcl 10625  ax-addrcl 10626  ax-mulcl 10627  ax-mulrcl 10628  ax-mulcom 10629  ax-addass 10630  ax-mulass 10631  ax-distr 10632  ax-i2m1 10633  ax-1ne0 10634  ax-1rid 10635  ax-rnegex 10636  ax-rrecex 10637  ax-cnre 10638  ax-pre-lttri 10639  ax-pre-lttrn 10640  ax-pre-ltadd 10641  ax-pre-mulgt0 10642  ax-pre-sup 10643

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-tp 4525  df-op 4527  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7578  df-1st 7691  df-2nd 7692  df-wrecs 7955  df-recs 8016  df-rdg 8054  df-er 8297  df-map 8416  df-en 8526  df-dom 8527  df-sdom 8528  df-sup 8929  df-inf 8930  df-pnf 10705  df-mnf 10706  df-xr 10707  df-ltxr 10708  df-le 10709  df-sub 10900  df-neg 10901  df-div 11326  df-nn 11665  df-2 11727  df-3 11728  df-n0 11925  df-z 12011  df-uz 12273  df-q 12379  df-rp 12421  df-xneg 12538  df-xadd 12539  df-xmul 12540  df-icc 12776  df-seq 13409  df-exp 13470  df-cj 14496  df-re 14497  df-im 14498  df-sqrt 14632  df-abs 14633  df-topgen 16765  df-psmet 20148  df-xmet 20149  df-met 20150  df-bl 20151  df-mopn 20152  df-top 21584  df-topon 21601  df-bases 21636  df-cn 21917  df-ii 23568  df-pconn 32689

This theorem is referenced by:  qtoppconn  32704