Theorem cnt1 23244

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 23134	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	3ad2ant3 1135	. 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 23140	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	5	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
7	6	ffnd 6692	. . . . . . 7 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 Fn ∪ 𝐽)
8		fnsnfv 6943	. . . . . . 7 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
9	7, 8	sylan 580	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
10	9	imaeq2d 6034	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ (𝐹 “ {𝑥})))
11		simpl2 1193	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹:𝑋–1-1→𝑌)
12	6	fdmd 6701	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom 𝐹 = ∪ 𝐽)
13		f1dm 6763	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1→𝑌 → dom 𝐹 = 𝑋)
14	13	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom 𝐹 = 𝑋)
15	12, 14	eqtr3d 2767	. . . . . . . . 9 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∪ 𝐽 = 𝑋)
16	15	eleq2d 2815	. . . . . . . 8 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ ∪ 𝐽 ↔ 𝑥 ∈ 𝑋))
17	16	biimpa 476	. . . . . . 7 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ 𝑋)
18	17	snssd 4776	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ⊆ 𝑋)
19		f1imacnv 6819	. . . . . 6 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ {𝑥} ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ {𝑥})) = {𝑥})
20	11, 18, 19	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ (𝐹 “ {𝑥})) = {𝑥})
21	10, 20	eqtrd 2765	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) = {𝑥})
22		simpl3 1194	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹 ∈ (𝐽 Cn 𝐾))
23		simpl1 1192	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐾 ∈ Fre)
24	6	ffvelcdmda 7059	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (𝐹‘𝑥) ∈ ∪ 𝐾)
25	4	t1sncld 23220	. . . . . 6 ⊢ ((𝐾 ∈ Fre ∧ (𝐹‘𝑥) ∈ ∪ 𝐾) → {(𝐹‘𝑥)} ∈ (Clsd‘𝐾))
26	23, 24, 25	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {(𝐹‘𝑥)} ∈ (Clsd‘𝐾))
27		cnclima 23162	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ {(𝐹‘𝑥)} ∈ (Clsd‘𝐾)) → (◡𝐹 “ {(𝐹‘𝑥)}) ∈ (Clsd‘𝐽))
28	22, 26, 27	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) ∈ (Clsd‘𝐽))
29	21, 28	eqeltrrd 2830	. . 3 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ∈ (Clsd‘𝐽))
30	29	ralrimiva 3126	. 2 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽))
31	3	ist1 23215	. 2 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽)))
32	2, 30, 31	sylanbrc 583	1 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Fre)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ⊆ wss 3917  {csn 4592  ∪ cuni 4874  ^◡ccnv 5640  dom cdm 5641   “ cima 5644   Fn wfn 6509  ⟶wf 6510  –1-1→wf1 6511  ‘cfv 6514  (class class class)co 7390  Topctop 22787  Clsdccld 22910   Cn ccn 23118  Frect1 23201

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-top 22788  df-topon 22805  df-cld 22913  df-cn 23121  df-t1 23208

This theorem is referenced by:  restt1  23261  sst1  23268  t1hmph  23685