Theorem cnt1 23273

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 23163	. . 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 2734	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2734	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 23169	. . . . . . . . 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 6703	. . . . . . 7 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 Fn ∪ 𝐽)
8		fnsnfv 6954	. . . . . . 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 6044	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ (𝐹 “ {𝑥})))
11		simpl2 1192	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹:𝑋–1-1→𝑌)
12	6	fdmd 6712	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom 𝐹 = ∪ 𝐽)
13		f1dm 6774	. . . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∪ 𝐽 = 𝑋)
16	15	eleq2d 2819	. . . . . . . 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 4782	. . . . . 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 584	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ (𝐹 “ {𝑥})) = {𝑥})
21	10, 20	eqtrd 2769	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (◡𝐹 “ {(𝐹‘𝑥)}) = {𝑥})
22		simpl3 1193	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹 ∈ (𝐽 Cn 𝐾))
23		simpl1 1191	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐾 ∈ Fre)
24	6	ffvelcdmda 7070	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → (𝐹‘𝑥) ∈ ∪ 𝐾)
25	4	t1sncld 23249	. . . . . 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 23191	. . . . 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 2834	. . 3 ⊢ (((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ∈ (Clsd‘𝐽))
30	29	ralrimiva 3130	. 2 ⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ ∪ 𝐽{𝑥} ∈ (Clsd‘𝐽))
31	3	ist1 23244	. 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 1539   ∈ wcel 2107  ∀wral 3050   ⊆ wss 3924  {csn 4599  ∪ cuni 4880  ^◡ccnv 5650  dom cdm 5651   “ cima 5654   Fn wfn 6522  ⟶wf 6523  –1-1→wf1 6524  ‘cfv 6527  (class class class)co 7399  Topctop 22816  Clsdccld 22939   Cn ccn 23147  Frect1 23230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7402  df-oprab 7403  df-mpo 7404  df-map 8836  df-top 22817  df-topon 22834  df-cld 22942  df-cn 23150  df-t1 23237

This theorem is referenced by:  restt1  23290  sst1  23297  t1hmph  23714