Theorem dnsconst 23272

Description: If a continuous mapping to a T₁ space is constant on a dense subset, it is constant on the entire space. Note that ((cls‘𝐽)‘𝐴) = 𝑋 means "𝐴 is dense in 𝑋 " and 𝐴 ⊆ (^◡𝐹 “ {𝑃}) means "𝐹 is constant on 𝐴 " (see funconstss 7031). (Contributed by NM, 15-Mar-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Hypotheses

Ref	Expression
dnsconst.1	⊢ 𝑋 = ∪ 𝐽
dnsconst.2	⊢ 𝑌 = ∪ 𝐾

Assertion

Ref	Expression
dnsconst	⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹:𝑋⟶{𝑃})

Proof of Theorem dnsconst

Step	Hyp	Ref	Expression
1		simplr 768	. . 3 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹 ∈ (𝐽 Cn 𝐾))
2		dnsconst.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3		dnsconst.2	. . . 4 ⊢ 𝑌 = ∪ 𝐾
4	2, 3	cnf 23140	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌)
5		ffn 6691	. . 3 ⊢ (𝐹:𝑋⟶𝑌 → 𝐹 Fn 𝑋)
6	1, 4, 5	3syl 18	. 2 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹 Fn 𝑋)
7		simpr3 1197	. . 3 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) = 𝑋)
8		simpll 766	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐾 ∈ Fre)
9		simpr1 1195	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝑃 ∈ 𝑌)
10	3	t1sncld 23220	. . . . . 6 ⊢ ((𝐾 ∈ Fre ∧ 𝑃 ∈ 𝑌) → {𝑃} ∈ (Clsd‘𝐾))
11	8, 9, 10	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → {𝑃} ∈ (Clsd‘𝐾))
12		cnclima 23162	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ {𝑃} ∈ (Clsd‘𝐾)) → (◡𝐹 “ {𝑃}) ∈ (Clsd‘𝐽))
13	1, 11, 12	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (◡𝐹 “ {𝑃}) ∈ (Clsd‘𝐽))
14		simpr2 1196	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ⊆ (◡𝐹 “ {𝑃}))
15	2	clsss2 22966	. . . 4 ⊢ (((◡𝐹 “ {𝑃}) ∈ (Clsd‘𝐽) ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃})) → ((cls‘𝐽)‘𝐴) ⊆ (◡𝐹 “ {𝑃}))
16	13, 14, 15	syl2anc 584	. . 3 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) ⊆ (◡𝐹 “ {𝑃}))
17	7, 16	eqsstrrd 3985	. 2 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝑋 ⊆ (◡𝐹 “ {𝑃}))
18		fconst3 7190	. 2 ⊢ (𝐹:𝑋⟶{𝑃} ↔ (𝐹 Fn 𝑋 ∧ 𝑋 ⊆ (◡𝐹 “ {𝑃})))
19	6, 17, 18	sylanbrc 583	1 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹:𝑋⟶{𝑃})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3917  {csn 4592  ∪ cuni 4874  ^◡ccnv 5640   “ cima 5644   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Clsdccld 22910  clsccl 22912   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-rep 5237  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-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  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-int 4914  df-iun 4960  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-cls 22915  df-cn 23121  df-t1 23208

This theorem is referenced by:  ipasslem8  30773