Theorem cnt1 23615

Description: The preimage of a T₁ topology under an injective map is T₁. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
cnt1	⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Fre)

Proof of Theorem cnt1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntop1 23505	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	3ad2ant3 1153	. 2 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
3		eqid 2760	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2760	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 23511	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	5	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
7	6	ffnd 6699	. . . . . . 7 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 Fn ∪ 𝐽)
8		fnsnfv 6953	. . . . . . 7 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
9	7, 8	sylan 592	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
10	9	imaeq2d 6051	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ (𝐹 “ {𝑥})))
11		simpl2 1211	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹:𝑋–1-1→𝑌)
12	6	fdmd 6709	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom 𝐹 = ∪ 𝐽)
13		f1dm 6773	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1→𝑌 → dom 𝐹 = 𝑋)
14	13	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom 𝐹 = 𝑋)
15	12, 14	eqtr3d 2797	. . . . . . . . 9 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∪ 𝐽 = 𝑋)
16	15	eleq2d 2846	. . . . . . . 8 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ ∪ 𝐽 ↔ 𝑥 ∈ 𝑋))
17	16	biimpa 482	. . . . . . 7 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ 𝑋)
18	17	snssd 4747	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ⊆ 𝑋)
19		f1imacnv 6830	. . . . . 6 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ {𝑥} ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ {𝑥})) = {𝑥})
20	11, 18, 19	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ (𝐹 “ {𝑥})) = {𝑥})
21	10, 20	eqtrd 2795	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) = {𝑥})
22		simpl3 1212	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹 ∈ (𝐽 Cn 𝐾))
23		simpl1 1210	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐾 ∈ Fre)
24	6	ffvelcdmda 7073	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (𝐹‘𝑥) ∈ ∪ 𝐾)
25	4	t1sncld 23591	. . . . . 6 ⊢ ((𝐾 ∈ Fre ∧ (𝐹‘𝑥) ∈ ∪ 𝐾) → {(𝐹‘𝑥)} ∈ (Clsd‘𝐾))
26	23, 24, 25	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} ∈ (Clsd‘𝐾))
27		cnclima 23533	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ {(𝐹‘𝑥)} ∈ (Clsd‘𝐾)) → (◡𝐹 “ {(𝐹‘𝑥)}) ∈ (Clsd‘𝐽))
28	22, 26, 27	syl2anc 596	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) ∈ (Clsd‘𝐽))
29	21, 28	eqeltrrd 2861	. . 3 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ∈ (Clsd‘𝐽))
30	29	ralrimiva 3154	. 2 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽))
31	3	ist1 23586	. 2 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽)))
32	2, 30, 31	sylanbrc 595	1 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Fre)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  {csn 4584  ∪ cuni 4867  ^◡ccnv 5647  dom cdm 5648   “ cima 5651   Fn wfn 6523  ⟶wf 6524  –1-1→wf1 6525  ‘cfv 6528  (class class class)co 7409  Topctop 23158  Clsdccld 23281   Cn ccn 23489  Frect1 23572

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 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8828  df-top 23159  df-topon 23176  df-cld 23284  df-cn 23492  df-t1 23579

This theorem is used by:  restt1  23632  sst1  23639  t1hmph  24057