Theorem cnpconn 33092

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 22300	. . 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 2738	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	pconncn 33086	. . . . . . . 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 767	. . . . . . . 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 22325	. . . . . . . 8 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ 𝑔) ∈ (II Cn 𝐾))
10	7, 8, 9	syl2anc 583	. . . . . . 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 23950	. . . . . . . . . . 11 ⊢ (0[,]1) = ∪ II
12	11, 3	cnf 22305	. . . . . . . . . 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 13130	. . . . . . . . 9 ⊢ 0 ∈ (0[,]1)
15		fvco3 6849	. . . . . . . . 9 ⊢ ((𝑔:(0[,]1)⟶∪ 𝐽 ∧ 0 ∈ (0[,]1)) → ((𝐹 ∘ 𝑔)‘0) = (𝐹‘(𝑔‘0)))
16	13, 14, 15	sylancl 585	. . . . . . . 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 777	. . . . . . . . 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 6760	. . . . . . . 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 2778	. . . . . . 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 13131	. . . . . . . . 9 ⊢ 1 ∈ (0[,]1)
21		fvco3 6849	. . . . . . . . 9 ⊢ ((𝑔:(0[,]1)⟶∪ 𝐽 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝑔)‘1) = (𝐹‘(𝑔‘1)))
22	13, 20, 21	sylancl 585	. . . . . . . 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 778	. . . . . . . . 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 6760	. . . . . . . 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 2778	. . . . . . 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 6755	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘0) = ((𝐹 ∘ 𝑔)‘0))
27	26	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘0) = (𝐹‘𝑢) ↔ ((𝐹 ∘ 𝑔)‘0) = (𝐹‘𝑢)))
28		fveq1 6755	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘1) = ((𝐹 ∘ 𝑔)‘1))
29	28	eqeq1d 2740	. . . . . . . . 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 3552	. . . . . . 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 833	. . . . . 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 3219	. . . . 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 3114	. . . 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 22305	. . . . . . . 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 6675	. . . . . . . 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 6676	. . . . . . 7 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹:∪ 𝐽⟶𝑌 ∧ ran 𝐹 = 𝑌))
41	37, 39, 40	sylanbrc 582	. . . . . 6 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽–onto→𝑌)
42		eqeq2 2750	. . . . . . . . 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 3225	. . . . . . 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 7141	. . . . . 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 3120	. . . 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 2750	. . . . . . . 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 3225	. . . . . 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 3120	. . . . 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 7141	. . . 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 33085	. 2 ⊢ (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
57	2, 55, 56	sylanbrc 582	1 ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  ∪ cuni 4836  ran crn 5581   ∘ ccom 5584  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255  0cc0 10802  1c1 10803  [,]cicc 13011  Topctop 21950   Cn ccn 22283  IIcii 23944  PConncpconn 33081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-icc 13015  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-topgen 17071  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-top 21951  df-topon 21968  df-bases 22004  df-cn 22286  df-ii 23946  df-pconn 33083

This theorem is referenced by:  qtoppconn  33098