Theorem cnt1 23076

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 22966	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	3ad2ant3 1133	. 2 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
3		eqid 2730	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2730	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 22972	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	5	3ad2ant3 1133	. . . . . . . 8 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
7	6	ffnd 6719	. . . . . . 7 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 Fn ∪ 𝐽)
8		fnsnfv 6971	. . . . . . 7 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
9	7, 8	sylan 578	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
10	9	imaeq2d 6060	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ (𝐹 “ {𝑥})))
11		simpl2 1190	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹:𝑋–1-1→𝑌)
12	6	fdmd 6729	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom 𝐹 = ∪ 𝐽)
13		f1dm 6792	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1→𝑌 → dom 𝐹 = 𝑋)
14	13	3ad2ant2 1132	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom 𝐹 = 𝑋)
15	12, 14	eqtr3d 2772	. . . . . . . . 9 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∪ 𝐽 = 𝑋)
16	15	eleq2d 2817	. . . . . . . 8 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ ∪ 𝐽 ↔ 𝑥 ∈ 𝑋))
17	16	biimpa 475	. . . . . . 7 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ 𝑋)
18	17	snssd 4813	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ⊆ 𝑋)
19		f1imacnv 6850	. . . . . 6 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ {𝑥} ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ {𝑥})) = {𝑥})
20	11, 18, 19	syl2anc 582	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ (𝐹 “ {𝑥})) = {𝑥})
21	10, 20	eqtrd 2770	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) = {𝑥})
22		simpl3 1191	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹 ∈ (𝐽 Cn 𝐾))
23		simpl1 1189	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐾 ∈ Fre)
24	6	ffvelcdmda 7087	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (𝐹‘𝑥) ∈ ∪ 𝐾)
25	4	t1sncld 23052	. . . . . 6 ⊢ ((𝐾 ∈ Fre ∧ (𝐹‘𝑥) ∈ ∪ 𝐾) → {(𝐹‘𝑥)} ∈ (Clsd‘𝐾))
26	23, 24, 25	syl2anc 582	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} ∈ (Clsd‘𝐾))
27		cnclima 22994	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ {(𝐹‘𝑥)} ∈ (Clsd‘𝐾)) → (◡𝐹 “ {(𝐹‘𝑥)}) ∈ (Clsd‘𝐽))
28	22, 26, 27	syl2anc 582	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) ∈ (Clsd‘𝐽))
29	21, 28	eqeltrrd 2832	. . 3 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ∈ (Clsd‘𝐽))
30	29	ralrimiva 3144	. 2 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽))
31	3	ist1 23047	. 2 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽)))
32	2, 30, 31	sylanbrc 581	1 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Fre)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ∀wral 3059   ⊆ wss 3949  {csn 4629  ∪ cuni 4909  ^◡ccnv 5676  dom cdm 5677   “ cima 5680   Fn wfn 6539  ⟶wf 6540  –1-1→wf1 6541  ‘cfv 6544  (class class class)co 7413  Topctop 22617  Clsdccld 22742   Cn ccn 22950  Frect1 23033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-map 8826  df-top 22618  df-topon 22635  df-cld 22745  df-cn 22953  df-t1 23040

This theorem is referenced by:  restt1  23093  sst1  23100  t1hmph  23517