Theorem cnpconn 35812

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 23469	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2	1	3ad2ant3 1153	. 2 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3		eqid 2760	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	pconncn 35806	. . . . . . . 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–onto→𝑌 ∧ 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–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 1233	. . . . . . . 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 23494	. . . . . . . 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–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 25112	. . . . . . . . . . 11 ⊢ (0[,]1) = ∪ II
12	11, 3	cnf 23474	. . . . . . . . . 10 ⊢ (𝑔 ∈ (II Cn 𝐽) → 𝑔:(0[,]1)⟶∪ 𝐽)
13	7, 12	syl 18	. . . . . . . . 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 13525	. . . . . . . . 9 ⊢ 0 ∈ (0[,]1)
15		fvco3 6979	. . . . . . . . 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–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 793	. . . . . . . . 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 6883	. . . . . . . 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 2795	. . . . . . 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 13526	. . . . . . . . 9 ⊢ 1 ∈ (0[,]1)
21		fvco3 6979	. . . . . . . . 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–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 794	. . . . . . . . 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 6883	. . . . . . . 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 2795	. . . . . . 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 6878	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘0) = ((𝐹 ∘ 𝑔)‘0))
27	26	eqeq1d 2762	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘0) = (𝐹‘𝑢) ↔ ((𝐹 ∘ 𝑔)‘0) = (𝐹‘𝑢)))
28		fveq1 6878	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘1) = ((𝐹 ∘ 𝑔)‘1))
29	28	eqeq1d 2762	. . . . . . . . 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 3576	. . . . . . 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–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 3169	. . . . 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 3205	. . . 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 23474	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
37	36	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌)
38		forn 6793	. . . . . . . 8 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
39	38	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = 𝑌)
40		dffo2 6794	. . . . . . 7 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹:∪ 𝐽⟶𝑌 ∧ ran 𝐹 = 𝑌))
41	37, 39, 40	sylanbrc 595	. . . . . 6 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽–onto→𝑌)
42		eqeq2 2772	. . . . . . . . 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 3186	. . . . . . 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 7291	. . . . . 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 18	. . . . 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 3185	. . . 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 2772	. . . . . . . 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 3186	. . . . . 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 3185	. . . . 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 7291	. . . 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 18	. . 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 35805	. 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–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  ∪ cuni 4867  ran crn 5656   ∘ ccom 5659  ⟶wf 6529  –onto→wfo 6531  ‘cfv 6533  (class class class)co 7414  0cc0 11127  1c1 11128  [,]cicc 13404  Topctop 23121   Cn ccn 23452  IIcii 25106  PConncpconn 35801

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-sup 9415  df-inf 9416  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-n0 12532  df-z 12619  df-uz 12891  df-q 13001  df-rp 13046  df-xneg 13166  df-xadd 13167  df-xmul 13168  df-icc 13408  df-seq 14069  df-exp 14129  df-cj 15189  df-re 15190  df-im 15191  df-sqrt 15325  df-abs 15326  df-topgen 17531  df-psmet 21580  df-xmet 21581  df-met 21582  df-bl 21583  df-mopn 21584  df-top 23122  df-topon 23139  df-bases 23174  df-cn 23455  df-ii 25108  df-pconn 35803

This theorem is used by:  qtoppconn  35818