Theorem cnpconn 36161

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⟶onto𝑌 ∧ 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 23586	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2	1	3ad2ant3 1153	. 2 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3		eqid 2763	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	pconncn 36155	. . . . . . . 8 ⊢ ((𝐽 ∈ PConn ∧ 𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
5	4	3expb 1138	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
6	5	3ad2antl1 1204	. . . . . 6 ⊢ (((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
7		simprl 783	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → 𝑔 ∈ (II Cn 𝐽))
8		simpll3 1233	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → 𝐹 ∈ (𝐽 Cn 𝐾))
9		cnco 23611	. . . . . . . 8 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ 𝑔) ∈ (II Cn 𝐾))
10	7, 8, 9	syl2anc 596	. . . . . . 7 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 25229	. . . . . . . . . . 11 ⊢ (0[,]1) = ∪ II
12	11, 3	cnf 23591	. . . . . . . . . 10 ⊢ (𝑔 ∈ (II Cn 𝐽) → 𝑔:(0[,]1)⟶∪ 𝐽)
13	7, 12	syl 18	. . . . . . . . 9 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → 𝑔:(0[,]1)⟶∪ 𝐽)
14		0elunit 13624	. . . . . . . . 9 ⊢ 0 ∈ (0[,]1)
15		fvco3 7005	. . . . . . . . 9 ⊢ ((𝑔:(0[,]1)⟶∪ 𝐽 ∧ 0 ∈ (0[,]1)) → ((𝐹 ∘ 𝑔)‘0) = (𝐹‘(𝑔‘0)))
16	13, 14, 15	sylancl 598	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 793	. . . . . . . . 9 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝑔‘0) = 𝑢)
18	17	fveq2d 6909	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 2798	. . . . . . 7 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 13625	. . . . . . . . 9 ⊢ 1 ∈ (0[,]1)
21		fvco3 7005	. . . . . . . . 9 ⊢ ((𝑔:(0[,]1)⟶∪ 𝐽 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝑔)‘1) = (𝐹‘(𝑔‘1)))
22	13, 20, 21	sylancl 598	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 794	. . . . . . . . 9 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝑔‘1) = 𝑣)
24	23	fveq2d 6909	. . . . . . . 8 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 2798	. . . . . . 7 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 6904	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘0) = ((𝐹 ∘ 𝑔)‘0))
27	26	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘0) = (𝐹‘𝑢) ↔ ((𝐹 ∘ 𝑔)‘0) = (𝐹‘𝑢)))
28		fveq1 6904	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘1) = ((𝐹 ∘ 𝑔)‘1))
29	28	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘1) = (𝐹‘𝑣) ↔ ((𝐹 ∘ 𝑔)‘1) = (𝐹‘𝑣)))
30	27, 29	anbi12d 644	. . . . . . . 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 3580	. . . . . . 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 850	. . . . . 6 ⊢ ((((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 3173	. . . . 5 ⊢ (((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ∈ ∪ 𝐽 ∧ 𝑣 ∈ ∪ 𝐽)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)))
34	33	ralrimivva 3209	. . . 4 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 23591	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
37	36	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌)
38		forn 6818	. . . . . . . 8 ⊢ (𝐹:𝑋⟶onto𝑌 → ran 𝐹 = 𝑌)
39	38	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = 𝑌)
40		dffo2 6819	. . . . . . 7 ⊢ (𝐹:∪ 𝐽⟶onto𝑌 ↔ (𝐹:∪ 𝐽⟶𝑌 ∧ ran 𝐹 = 𝑌))
41	37, 39, 40	sylanbrc 595	. . . . . 6 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶onto𝑌)
42		eqeq2 2775	. . . . . . . . 9 ⊢ ((𝐹‘𝑣) = 𝑦 → ((𝑓‘1) = (𝐹‘𝑣) ↔ (𝑓‘1) = 𝑦))
43	42	anbi2d 642	. . . . . . . 8 ⊢ ((𝐹‘𝑣) = 𝑦 → (((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = (𝐹‘𝑣)) ↔ ((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦)))
44	43	rexbidv 3190	. . . . . . 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 7317	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶onto𝑌 → (∀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 18	. . . . 5 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 3189	. . . 4 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 ∈ ∪ 𝐽∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦))
49		eqeq2 2775	. . . . . . . 8 ⊢ ((𝐹‘𝑢) = 𝑥 → ((𝑓‘0) = (𝐹‘𝑢) ↔ (𝑓‘0) = 𝑥))
50	49	anbi1d 643	. . . . . . 7 ⊢ ((𝐹‘𝑢) = 𝑥 → (((𝑓‘0) = (𝐹‘𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
51	50	rexbidv 3190	. . . . . 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 3189	. . . . 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 7317	. . . 4 ⊢ (𝐹:∪ 𝐽⟶onto𝑌 → (∀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 18	. . 3 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
56	35	ispconn 36154	. 2 ⊢ (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
57	2, 55, 56	sylanbrc 595	1 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ∪ cuni 4870  ran crn 5656   ∘ ccom 5659  ⟶wf 6551  ⟶ontowfo 6553  ‘cfv 6555  (class class class)co 7440  0cc0 11224  1c1 11225  [,]cicc 13503  Topctop 23238   Cn ccn 23569  IIcii 25223  PConncpconn 36150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-sup 9458  df-inf 9459  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-z 12718  df-uz 12990  df-q 13100  df-rp 13145  df-xneg 13265  df-xadd 13266  df-xmul 13267  df-icc 13507  df-seq 14169  df-exp 14229  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-abs 15429  df-topgen 17640  df-psmet 21697  df-xmet 21698  df-met 21699  df-bl 21700  df-mopn 21701  df-top 23239  df-topon 23256  df-bases 23291  df-cn 23572  df-ii 25225  df-pconn 36152

This theorem is used by:  qtoppconn  36167