Theorem cnpconn 34873

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 23165	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2	1	3ad2ant3 1132	. 2 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3		eqid 2728	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	pconncn 34867	. . . . . . . 8 ⊢ ((𝐽 ∈ PConn ∧ 𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
5	4	3expb 1117	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
6	5	3ad2antl1 1182	. . . . . 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 769	. . . . . . . 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 1211	. . . . . . . 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 23190	. . . . . . . 8 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ 𝑔) ∈ (II Cn 𝐾))
10	7, 8, 9	syl2anc 582	. . . . . . 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 24821	. . . . . . . . . . 11 ⊢ (0[,]1) = ∪ II
12	11, 3	cnf 23170	. . . . . . . . . 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 13486	. . . . . . . . 9 ⊢ 0 ∈ (0[,]1)
15		fvco3 7002	. . . . . . . . 9 ⊢ ((𝑔:(0[,]1)⟶∪ 𝐽 ∧ 0 ∈ (0[,]1)) → ((𝐹 ∘ 𝑔)‘0) = (𝐹‘(𝑔‘0)))
16	13, 14, 15	sylancl 584	. . . . . . . 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 779	. . . . . . . . 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 6906	. . . . . . . 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 2768	. . . . . . 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 13487	. . . . . . . . 9 ⊢ 1 ∈ (0[,]1)
21		fvco3 7002	. . . . . . . . 9 ⊢ ((𝑔:(0[,]1)⟶∪ 𝐽 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝑔)‘1) = (𝐹‘(𝑔‘1)))
22	13, 20, 21	sylancl 584	. . . . . . . 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 780	. . . . . . . . 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 6906	. . . . . . . 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 2768	. . . . . . 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 6901	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘0) = ((𝐹 ∘ 𝑔)‘0))
27	26	eqeq1d 2730	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘0) = (𝐹‘𝑢) ↔ ((𝐹 ∘ 𝑔)‘0) = (𝐹‘𝑢)))
28		fveq1 6901	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘1) = ((𝐹 ∘ 𝑔)‘1))
29	28	eqeq1d 2730	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘1) = (𝐹‘𝑣) ↔ ((𝐹 ∘ 𝑔)‘1) = (𝐹‘𝑣)))
30	27, 29	anbi12d 630	. . . . . . . 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 3611	. . . . . . 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 835	. . . . . 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 3158	. . . . 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 3198	. . . 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 23170	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
37	36	3ad2ant3 1132	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌)
38		forn 6819	. . . . . . . 8 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
39	38	3ad2ant2 1131	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = 𝑌)
40		dffo2 6820	. . . . . . 7 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹:∪ 𝐽⟶𝑌 ∧ ran 𝐹 = 𝑌))
41	37, 39, 40	sylanbrc 581	. . . . . 6 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽–onto→𝑌)
42		eqeq2 2740	. . . . . . . . 9 ⊢ ((𝐹‘𝑣) = 𝑦 → ((𝑓‘1) = (𝐹‘𝑣) ↔ (𝑓‘1) = 𝑦))
43	42	anbi2d 628	. . . . . . . 8 ⊢ ((𝐹‘𝑣) = 𝑦 → (((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)) ↔ ((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦)))
44	43	rexbidv 3176	. . . . . . 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 7304	. . . . . 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 3175	. . . 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 231	. . 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 2740	. . . . . . . 8 ⊢ ((𝐹‘𝑢) = 𝑥 → ((𝑓‘0) = (𝐹‘𝑢) ↔ (𝑓‘0) = 𝑥))
50	49	anbi1d 629	. . . . . . 7 ⊢ ((𝐹‘𝑢) = 𝑥 → (((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
51	50	rexbidv 3176	. . . . . 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 3175	. . . . 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 7304	. . . 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 231	. 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 34866	. 2 ⊢ (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
57	2, 55, 56	sylanbrc 581	1 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ∃wrex 3067  ∪ cuni 4912  ran crn 5683   ∘ ccom 5686  ⟶wf 6549  –onto→wfo 6551  ‘cfv 6553  (class class class)co 7426  0cc0 11146  1c1 11147  [,]cicc 13367  Topctop 22815   Cn ccn 23148  IIcii 24815  PConncpconn 34862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-sup 9473  df-inf 9474  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-q 12971  df-rp 13015  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-icc 13371  df-seq 14007  df-exp 14067  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-topgen 17432  df-psmet 21278  df-xmet 21279  df-met 21280  df-bl 21281  df-mopn 21282  df-top 22816  df-topon 22833  df-bases 22869  df-cn 23151  df-ii 24817  df-pconn 34864

This theorem is referenced by:  qtoppconn  34879