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Theorem cnhaus 23341

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 23227	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	3ad2ant3 1142	. 2 ⊢ ((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
3		simpl1 1199	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝐾 ∈ Haus)
4		simpl3 1201	. . . . . . . 8 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝐹 ∈ (𝐽 Cn 𝐾))
5		eqid 2741	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
6		eqid 2741	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ 𝐾
7	5, 6	cnf 23233	. . . . . . . 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 785	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ ∪ 𝐽)
10	8, 9	ffvelcdmd 7030	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ∈ ∪ 𝐾)
11		simprlr 786	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ ∪ 𝐽)
12	8, 11	ffvelcdmd 7030	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑦) ∈ ∪ 𝐾)
13		simprr 779	. . . . . . 7 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
14		simpl2 1200	. . . . . . . . 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 6669	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → dom 𝐹 = ∪ 𝐽)
16		f1dm 6731	. . . . . . . . . . . 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 2778	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → ∪ 𝐽 = 𝑋)
19	9, 18	eleqtrd 2843	. . . . . . . . 9 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝑋)
20	11, 18	eleqtrd 2843	. . . . . . . . 9 ⊢ (((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝑋)
21		f1fveq 7210	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
22	14, 19, 20, 21	syl12anc 843	. . . . . . . 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 2980	. . . . . . 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 259	. . . . . 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 23315	. . . . . 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 1381	. . . . 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 1222	. . . . . . . . 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 785	. . . . . . . . 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 23252	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑢 ∈ 𝐾) → (^◡𝐹 “ 𝑢) ∈ 𝐽)
30	27, 28, 29	syl2anc 591	. . . . . . . 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 786	. . . . . . . . 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 23252	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑣 ∈ 𝐾) → (^◡𝐹 “ 𝑣) ∈ 𝐽)
33	27, 31, 32	syl2anc 591	. . . . . . . 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 482	. . . . . . . . 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 1229	. . . . . . . . 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 482	. . . . . . . . . . 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 6660	. . . . . . . . . 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 7003	. . . . . . . . . 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 720	. . . . . . . 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 482	. . . . . . . . 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 1230	. . . . . . . . 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 7003	. . . . . . . . . 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 720	. . . . . . . 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 6662	. . . . . . . . . 10 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Fun 𝐹)
47		inpreima 7009	. . . . . . . . . 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 1231	. . . . . . . . . . 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 6019	. . . . . . . . . 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 6036	. . . . . . . . . 10 ⊢ (^◡𝐹 “ ∅) = ∅
52	50, 51	eqtrdi 2792	. . . . . . . . 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 2778	. . . . . . . 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 2830	. . . . . . . . . 10 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → (𝑥 ∈ 𝑚 ↔ 𝑥 ∈ (^◡𝐹 “ 𝑢)))
55		ineq1 4145	. . . . . . . . . . 11 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → (𝑚 ∩ 𝑛) = ((^◡𝐹 “ 𝑢) ∩ 𝑛))
56	55	eqeq1d 2743	. . . . . . . . . 10 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → ((𝑚 ∩ 𝑛) = ∅ ↔ ((^◡𝐹 “ 𝑢) ∩ 𝑛) = ∅))
57	54, 56	3anbi13d 1447	. . . . . . . . 9 ⊢ (𝑚 = (^◡𝐹 “ 𝑢) → ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ (𝑥 ∈ (^◡𝐹 “ 𝑢) ∧ 𝑦 ∈ 𝑛 ∧ ((^◡𝐹 “ 𝑢) ∩ 𝑛) = ∅)))
58		eleq2 2830	. . . . . . . . . 10 ⊢ (𝑛 = (^◡𝐹 “ 𝑣) → (𝑦 ∈ 𝑛 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑣)))
59		ineq2 4146	. . . . . . . . . . 11 ⊢ (𝑛 = (^◡𝐹 “ 𝑣) → ((^◡𝐹 “ 𝑢) ∩ 𝑛) = ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)))
60	59	eqeq1d 2743	. . . . . . . . . 10 ⊢ (𝑛 = (^◡𝐹 “ 𝑣) → (((^◡𝐹 “ 𝑢) ∩ 𝑛) = ∅ ↔ ((^◡𝐹 “ 𝑢) ∩ (^◡𝐹 “ 𝑣)) = ∅))
61	58, 60	3anbi23d 1448	. . . . . . . . 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 3575	. . . . . . . 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 1391	. . . . . . 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 458	. . . . . 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 3198	. . . . 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 458	. . 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 3184	. 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 23309	. 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 590	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 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∀wral 3055 ∃wrex 3065 ∩ cin 3884 ∅c0 4264 ∪ cuni 4841 ^◡ccnv 5620 dom cdm 5621 “ cima 5624 Fun wfun 6483 Fn wfn 6484 ⟶wf 6485 –1-1→wf1 6486 ‘cfv 6489 (class class class)co 7360 Topctop 22880 Cn ccn 23211 Hauscha 23295

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-sbc 3726 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fv 6497 df-ov 7363 df-oprab 7364 df-mpo 7365 df-map 8769 df-top 22881 df-topon 22898 df-cn 23214 df-haus 23302

This theorem is referenced by: resthaus 23355 sshaus 23362 haushmph 23779

W3C validator