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Theorem cnhaus 21063

Description: The preimage of a Hausdorff topology under an injective map is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)

**Assertion**
Ref	Expression
cnhaus	⊢ ((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Haus)

Proof of Theorem cnhaus Dummy variables 𝑥 𝑦 𝑣 𝑢 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntop1 20949	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	3ad2ant3 1082	. 2 ⊢ ((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
3		simpl1 1062	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝐾 ∈ Haus)
4		simpl3 1064	. . . . . . . 8 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝐹 ∈ (𝐽 Cn 𝐾))
5		eqid 2626	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
6		eqid 2626	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ 𝐾
7	5, 6	cnf 20955	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
8	4, 7	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
9		simprll 801	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ ∪ 𝐽)
10	8, 9	ffvelrnd 6317	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ∈ ∪ 𝐾)
11		simprlr 802	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ ∪ 𝐽)
12	8, 11	ffvelrnd 6317	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑦) ∈ ∪ 𝐾)
13		simprr 795	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
14		simpl2 1063	. . . . . . . . 9 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝐹:𝑋–1-1→𝑌)
15		fdm 6010	. . . . . . . . . . . 12 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → dom 𝐹 = ∪ 𝐽)
16	8, 15	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → dom 𝐹 = ∪ 𝐽)
17		f1dm 6064	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–1-1→𝑌 → dom 𝐹 = 𝑋)
18	14, 17	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → dom 𝐹 = 𝑋)
19	16, 18	eqtr3d 2662	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → ∪ 𝐽 = 𝑋)
20	9, 19	eleqtrd 2706	. . . . . . . . 9 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝑋)
21	11, 19	eleqtrd 2706	. . . . . . . . 9 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝑋)
22		f1fveq 6474	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
23	14, 20, 21, 22	syl12anc 1321	. . . . . . . 8 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
24	23	necon3bid 2840	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ 𝑥 ≠ 𝑦))
25	13, 24	mpbird 247	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ≠ (𝐹‘𝑦))
26	6	hausnei 21037	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ ((𝐹‘𝑥) ∈ ∪ 𝐾 ∧ (𝐹‘𝑦) ∈ ∪ 𝐾 ∧ (𝐹‘𝑥) ≠ (𝐹‘𝑦))) → ∃𝑢 ∈ 𝐾 ∃𝑣 ∈ 𝐾 ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
27	3, 10, 12, 25, 26	syl13anc 1325	. . . . 5 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → ∃𝑢 ∈ 𝐾 ∃𝑣 ∈ 𝐾 ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
28		simpll3 1100	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝐹 ∈ (𝐽 Cn 𝐾))
29		simprll 801	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑢 ∈ 𝐾)
30		cnima 20974	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑢 ∈ 𝐾) → (^◡𝐹 “ 𝑢) ∈ 𝐽)
31	28, 29, 30	syl2anc 692	. . . . . . . 8 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (^◡𝐹 “ 𝑢) ∈ 𝐽)
32		simprlr 802	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑣 ∈ 𝐾)
33		cnima 20974	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑣 ∈ 𝐾) → (^◡𝐹 “ 𝑣) ∈ 𝐽)
34	28, 32, 33	syl2anc 692	. . . . . . . 8 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (^◡𝐹 “ 𝑣) ∈ 𝐽)
35	9	adantr 481	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑥 ∈ ∪ 𝐽)
36		simprr1 1107	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (𝐹‘𝑥) ∈ 𝑢)
37	8	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝐹:∪ 𝐽⟶∪ 𝐾)
38		ffn 6004	. . . . . . . . . . 11 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → 𝐹 Fn ∪ 𝐽)
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝐹 Fn ∪ 𝐽)
40		elpreima 6294	. . . . . . . . . 10 ⊢ (𝐹 Fn ∪ 𝐽 → (𝑥 ∈ (^◡𝐹 “ 𝑢) ↔ (𝑥 ∈ ∪ 𝐽 ∧ (𝐹‘𝑥) ∈ 𝑢)))
41	39, 40	syl 17	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (𝑥 ∈ (^◡𝐹 “ 𝑢) ↔ (𝑥 ∈ ∪ 𝐽 ∧ (𝐹‘𝑥) ∈ 𝑢)))
42	35, 36, 41	mpbir2and 956	. . . . . . . 8 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑥 ∈ (^◡𝐹 “ 𝑢))
43	11	adantr 481	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑦 ∈ ∪ 𝐽)
44		simprr2 1108	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (𝐹‘𝑦) ∈ 𝑣)
45		elpreima 6294	. . . . . . . . . 10 ⊢ (𝐹 Fn ∪ 𝐽 → (𝑦 ∈ (^◡𝐹 “ 𝑣) ↔ (𝑦 ∈ ∪ 𝐽 ∧ (𝐹‘𝑦) ∈ 𝑣)))
46	39, 45	syl 17	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (𝑦 ∈ (^◡𝐹 “ 𝑣) ↔ (𝑦 ∈ ∪ 𝐽 ∧ (𝐹‘𝑦) ∈ 𝑣)))
47	43, 44, 46	mpbir2and 956	. . . . . . . 8 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑦 ∈ (^◡𝐹 “ 𝑣))
48		ffun 6007	. . . . . . . . . 10 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Fun 𝐹)
49		inpreima 6299	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (^◡𝐹 “ (𝑢 ∩ 𝑣)) = ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)))
50	37, 48, 49	3syl 18	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (^◡𝐹 “ (𝑢 ∩ 𝑣)) = ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)))
51		simprr3 1109	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (𝑢 ∩ 𝑣) = ∅)
52	51	imaeq2d 5429	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (^◡𝐹 “ (𝑢 ∩ 𝑣)) = (^◡𝐹 “ ∅))
53		ima0 5444	. . . . . . . . . 10 ⊢ (^◡𝐹 “ ∅) = ∅
54	52, 53	syl6eq 2676	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (^◡𝐹 “ (𝑢 ∩ 𝑣)) = ∅)
55	50, 54	eqtr3d 2662	. . . . . . . 8 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)) = ∅)
56		eleq2 2693	. . . . . . . . . 10 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → (𝑥 ∈ 𝑚 ↔ 𝑥 ∈ (^◡𝐹 “ 𝑢)))
57		ineq1 3790	. . . . . . . . . . 11 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → (𝑚 ∩ 𝑛) = ((^◡𝐹 “ 𝑢) ∩ 𝑛))
58	57	eqeq1d 2628	. . . . . . . . . 10 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → ((𝑚 ∩ 𝑛) = ∅ ↔ ((^◡𝐹 “ 𝑢) ∩ 𝑛) = ∅))
59	56, 58	3anbi13d 1398	. . . . . . . . 9 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ (𝑥 ∈ (^◡𝐹 “ 𝑢) ∧ 𝑦 ∈ 𝑛 ∧ ((^◡𝐹 “ 𝑢) ∩ 𝑛) = ∅)))
60		eleq2 2693	. . . . . . . . . 10 ⊢ (𝑛 = (^◡𝐹 “ 𝑣) → (𝑦 ∈ 𝑛 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑣)))
61		ineq2 3791	. . . . . . . . . . 11 ⊢ (𝑛 = (^◡𝐹 “ 𝑣) → ((^◡𝐹 “ 𝑢) ∩ 𝑛) = ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)))
62	61	eqeq1d 2628	. . . . . . . . . 10 ⊢ (𝑛 = (^◡𝐹 “ 𝑣) → (((^◡𝐹 “ 𝑢) ∩ 𝑛) = ∅ ↔ ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)) = ∅))
63	60, 62	3anbi23d 1399	. . . . . . . . 9 ⊢ (𝑛 = (^◡𝐹 “ 𝑣) → ((𝑥 ∈ (^◡𝐹 “ 𝑢) ∧ 𝑦 ∈ 𝑛 ∧ ((^◡𝐹 “ 𝑢) ∩ 𝑛) = ∅) ↔ (𝑥 ∈ (^◡𝐹 “ 𝑢) ∧ 𝑦 ∈ (^◡𝐹 “ 𝑣) ∧ ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)) = ∅)))
64	59, 63	rspc2ev 3313	. . . . . . . 8 ⊢ (((^◡𝐹 “ 𝑢) ∈ 𝐽 ∧ (^◡𝐹 “ 𝑣) ∈ 𝐽 ∧ (𝑥 ∈ (^◡𝐹 “ 𝑢) ∧ 𝑦 ∈ (^◡𝐹 “ 𝑣) ∧ ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)) = ∅)) → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
65	31, 34, 42, 47, 55, 64	syl113anc 1335	. . . . . . 7 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ ((𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾) ∧ ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
66	65	expr 642	. . . . . 6 ⊢ ((((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) ∧ (𝑢 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾)) → (((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
67	66	rexlimdvva 3036	. . . . 5 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → (∃𝑢 ∈ 𝐾 ∃𝑣 ∈ 𝐾 ((𝐹‘𝑥) ∈ 𝑢 ∧ (𝐹‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
68	27, 67	mpd 15	. . . 4 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
69	68	expr 642	. . 3 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽)) → (𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
70	69	ralrimivva 2970	. 2 ⊢ ((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
71	5	ishaus 21031	. 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
72	2, 70, 71	sylanbrc 697	1 ⊢ ((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Haus)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1036 = wceq 1480 ∈ wcel 1992 ≠ wne 2796 ∀wral 2912 ∃wrex 2913 ∩ cin 3559 ∅c0 3896 ∪ cuni 4407 ^◡ccnv 5078 dom cdm 5079 “ cima 5082 Fun wfun 5844 Fn wfn 5845 ⟶wf 5846 –1-1→wf1 5847 ‘cfv 5850 (class class class)co 6605 Topctop 20612 Cn ccn 20933 Hauscha 21017

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1841 ax-6 1890 ax-7 1937 ax-8 1994 ax-9 2001 ax-10 2021 ax-11 2036 ax-12 2049 ax-13 2250 ax-ext 2606 ax-sep 4746 ax-nul 4754 ax-pow 4808 ax-pr 4872 ax-un 6903

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1038 df-tru 1483 df-ex 1702 df-nf 1707 df-sb 1883 df-eu 2478 df-mo 2479 df-clab 2613 df-cleq 2619 df-clel 2622 df-nfc 2756 df-ne 2797 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3193 df-sbc 3423 df-dif 3563 df-un 3565 df-in 3567 df-ss 3574 df-nul 3897 df-if 4064 df-pw 4137 df-sn 4154 df-pr 4156 df-op 4160 df-uni 4408 df-br 4619 df-opab 4679 df-mpt 4680 df-id 4994 df-xp 5085 df-rel 5086 df-cnv 5087 df-co 5088 df-dm 5089 df-rn 5090 df-res 5091 df-ima 5092 df-iota 5813 df-fun 5852 df-fn 5853 df-f 5854 df-f1 5855 df-fv 5858 df-ov 6608 df-oprab 6609 df-mpt2 6610 df-map 7805 df-top 20616 df-topon 20618 df-cn 20936 df-haus 21024

This theorem is referenced by: resthaus 21077 sshaus 21084 haushmph 21500

W3C validator