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Theorem cnhaus 23315

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 23201	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	3ad2ant3 1136	. 2 ⊢ ((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
3		simpl1 1193	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝐾 ∈ Haus)
4		simpl3 1195	. . . . . . . 8 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝐹 ∈ (𝐽 Cn 𝐾))
5		eqid 2737	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
6		eqid 2737	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ 𝐾
7	5, 6	cnf 23207	. . . . . . . 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 779	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ ∪ 𝐽)
10	8, 9	ffvelcdmd 7041	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ∈ ∪ 𝐾)
11		simprlr 780	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ ∪ 𝐽)
12	8, 11	ffvelcdmd 7041	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑦) ∈ ∪ 𝐾)
13		simprr 773	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
14		simpl2 1194	. . . . . . . . 9 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝐹:𝑋–1-1→𝑌)
15	8	fdmd 6682	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → dom 𝐹 = ∪ 𝐽)
16		f1dm 6744	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–1-1→𝑌 → dom 𝐹 = 𝑋)
17	14, 16	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → dom 𝐹 = 𝑋)
18	15, 17	eqtr3d 2774	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → ∪ 𝐽 = 𝑋)
19	9, 18	eleqtrd 2839	. . . . . . . . 9 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝑋)
20	11, 18	eleqtrd 2839	. . . . . . . . 9 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝑋)
21		f1fveq 7220	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
22	14, 19, 20, 21	syl12anc 837	. . . . . . . 8 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
23	22	necon3bid 2977	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ 𝑥 ≠ 𝑦))
24	13, 23	mpbird 257	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ≠ (𝐹‘𝑦))
25	6	hausnei 23289	. . . . . 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) = ∅))
26	3, 10, 12, 24, 25	syl13anc 1375	. . . . 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) = ∅))
27		simpll3 1216	. . . . . . . . 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))
28		simprll 779	. . . . . . . . 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)
29		cnima 23226	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑢 ∈ 𝐾) → (^◡𝐹 “ 𝑢) ∈ 𝐽)
30	27, 28, 29	syl2anc 585	. . . . . . . 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)
31		simprlr 780	. . . . . . . . 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)
32		cnima 23226	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑣 ∈ 𝐾) → (^◡𝐹 “ 𝑣) ∈ 𝐽)
33	27, 31, 32	syl2anc 585	. . . . . . . 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)
34	9	adantr 480	. . . . . . . . 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)
35		simprr1 1223	. . . . . . . . 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)
36	8	adantr 480	. . . . . . . . . . 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)
37	36	ffnd 6673	. . . . . . . . . 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)
38		elpreima 7014	. . . . . . . . . 10 ⊢ (𝐹 Fn ∪ 𝐽 → (𝑥 ∈ (^◡𝐹 “ 𝑢) ↔ (𝑥 ∈ ∪ 𝐽 ∧ (𝐹‘𝑥) ∈ 𝑢)))
39	37, 38	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)))
40	34, 35, 39	mpbir2and 714	. . . . . . . 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))
41	11	adantr 480	. . . . . . . . 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)
42		simprr2 1224	. . . . . . . . 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)
43		elpreima 7014	. . . . . . . . . 10 ⊢ (𝐹 Fn ∪ 𝐽 → (𝑦 ∈ (^◡𝐹 “ 𝑣) ↔ (𝑦 ∈ ∪ 𝐽 ∧ (𝐹‘𝑦) ∈ 𝑣)))
44	37, 43	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)))
45	41, 42, 44	mpbir2and 714	. . . . . . . 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))
46		ffun 6675	. . . . . . . . . 10 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Fun 𝐹)
47		inpreima 7020	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (^◡𝐹 “ (𝑢 ∩ 𝑣)) = ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)))
48	36, 46, 47	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)))
49		simprr3 1225	. . . . . . . . . . 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) = ∅)
50	49	imaeq2d 6029	. . . . . . . . . 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 “ ∅))
51		ima0 6046	. . . . . . . . . 10 ⊢ (^◡𝐹 “ ∅) = ∅
52	50, 51	eqtrdi 2788	. . . . . . . . 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)) = ∅)
53	48, 52	eqtr3d 2774	. . . . . . . 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)) = ∅)
54		eleq2 2826	. . . . . . . . . 10 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → (𝑥 ∈ 𝑚 ↔ 𝑥 ∈ (^◡𝐹 “ 𝑢)))
55		ineq1 4167	. . . . . . . . . . 11 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → (𝑚 ∩ 𝑛) = ((^◡𝐹 “ 𝑢) ∩ 𝑛))
56	55	eqeq1d 2739	. . . . . . . . . 10 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → ((𝑚 ∩ 𝑛) = ∅ ↔ ((^◡𝐹 “ 𝑢) ∩ 𝑛) = ∅))
57	54, 56	3anbi13d 1441	. . . . . . . . 9 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ (𝑥 ∈ (^◡𝐹 “ 𝑢) ∧ 𝑦 ∈ 𝑛 ∧ ((^◡𝐹 “ 𝑢) ∩ 𝑛) = ∅)))
58		eleq2 2826	. . . . . . . . . 10 ⊢ (𝑛 = (^◡𝐹 “ 𝑣) → (𝑦 ∈ 𝑛 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑣)))
59		ineq2 4168	. . . . . . . . . . 11 ⊢ (𝑛 = (^◡𝐹 “ 𝑣) → ((^◡𝐹 “ 𝑢) ∩ 𝑛) = ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)))
60	59	eqeq1d 2739	. . . . . . . . . 10 ⊢ (𝑛 = (^◡𝐹 “ 𝑣) → (((^◡𝐹 “ 𝑢) ∩ 𝑛) = ∅ ↔ ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)) = ∅))
61	58, 60	3anbi23d 1442	. . . . . . . . 9 ⊢ (𝑛 = (^◡𝐹 “ 𝑣) → ((𝑥 ∈ (^◡𝐹 “ 𝑢) ∧ 𝑦 ∈ 𝑛 ∧ ((^◡𝐹 “ 𝑢) ∩ 𝑛) = ∅) ↔ (𝑥 ∈ (^◡𝐹 “ 𝑢) ∧ 𝑦 ∈ (^◡𝐹 “ 𝑣) ∧ ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)) = ∅)))
62	57, 61	rspc2ev 3591	. . . . . . . 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) = ∅))
63	30, 33, 40, 45, 53, 62	syl113anc 1385	. . . . . . 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) = ∅))
64	63	expr 456	. . . . . 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) = ∅)))
65	64	rexlimdvva 3195	. . . . 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) = ∅)))
66	26, 65	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) = ∅))
67	66	expr 456	. . 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) = ∅)))
68	67	ralrimivva 3181	. 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) = ∅)))
69	5	ishaus 23283	. 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
70	2, 68, 69	sylanbrc 584	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ∩ cin 3902 ∅c0 4287 ∪ cuni 4865 ^◡ccnv 5633 dom cdm 5634 “ cima 5637 Fun wfun 6496 Fn wfn 6497 ⟶wf 6498 –1-1→wf1 6499 ‘cfv 6502 (class class class)co 7370 Topctop 22854 Cn ccn 23185 Hauscha 23269

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5529 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fv 6510 df-ov 7373 df-oprab 7374 df-mpo 7375 df-map 8779 df-top 22855 df-topon 22872 df-cn 23188 df-haus 23276

This theorem is referenced by: resthaus 23329 sshaus 23336 haushmph 23753

W3C validator