Theorem cnt1 23695

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⟶1-1𝑌 ∧ 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 23585	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	3ad2ant3 1153	. 2 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
3		eqid 2763	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2763	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 23591	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	5	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
7	6	ffnd 6729	. . . . . . 7 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 Fn ∪ 𝐽)
8		fnsnfv 6984	. . . . . . 7 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
9	7, 8	sylan 592	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
10	9	imaeq2d 6063	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ (𝐹 “ {𝑥})))
11		simpl2 1211	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹:𝑋⟶1-1𝑌)
12	6	fdmd 6739	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom 𝐹 = ∪ 𝐽)
13		f1dm 6803	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶1-1𝑌 → dom 𝐹 = 𝑋)
14	13	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom 𝐹 = 𝑋)
15	12, 14	eqtr3d 2800	. . . . . . . . 9 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∪ 𝐽 = 𝑋)
16	15	eleq2d 2850	. . . . . . . 8 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ ∪ 𝐽 ↔ 𝑥 ∈ 𝑋))
17	16	biimpa 482	. . . . . . 7 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ 𝑋)
18	17	snssd 4747	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ⊆ 𝑋)
19		f1imacnv 6860	. . . . . 6 ⊢ ((𝐹:𝑋⟶1-1𝑌 ∧ {𝑥} ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ {𝑥})) = {𝑥})
20	11, 18, 19	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ (𝐹 “ {𝑥})) = {𝑥})
21	10, 20	eqtrd 2798	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) = {𝑥})
22		simpl3 1212	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹 ∈ (𝐽 Cn 𝐾))
23		simpl1 1210	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐾 ∈ Fre)
24	6	ffvelcdmda 7105	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (𝐹‘𝑥) ∈ ∪ 𝐾)
25	4	t1sncld 23671	. . . . . 6 ⊢ ((𝐾 ∈ Fre ∧ (𝐹‘𝑥) ∈ ∪ 𝐾) → {(𝐹‘𝑥)} ∈ (Clsd‘𝐾))
26	23, 24, 25	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} ∈ (Clsd‘𝐾))
27		cnclima 23613	. . . . 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⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) ∈ (Clsd‘𝐽))
29	21, 28	eqeltrrd 2865	. . 3 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ∈ (Clsd‘𝐽))
30	29	ralrimiva 3158	. 2 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽))
31	3	ist1 23666	. 2 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽)))
32	2, 30, 31	sylanbrc 595	1 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋⟶1-1𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Fre)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  {csn 4588  ∪ cuni 4870  ^◡ccnv 5654  dom cdm 5655   “ cima 5658   Fn wfn 6550  ⟶wf 6551  ⟶1-1wf1 6552  ‘cfv 6555  (class class class)co 7440  Topctop 23238  Clsdccld 23361   Cn ccn 23569  Frect1 23652

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-map 8874  df-top 23239  df-topon 23256  df-cld 23364  df-cn 23572  df-t1 23659

This theorem is used by:  restt1  23712  sst1  23719  t1hmph  24137