Theorem dnsconst 22881

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 7057). (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 767	. . 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 22749	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌)
5		ffn 6717	. . 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 1196	. . 3 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) = 𝑋)
8		simpll 765	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐾 ∈ Fre)
9		simpr1 1194	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝑃 ∈ 𝑌)
10	3	t1sncld 22829	. . . . . 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 22771	. . . . 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 1195	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ⊆ (◡𝐹 “ {𝑃}))
15	2	clsss2 22575	. . . 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 4021	. 2 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝑋 ⊆ (◡𝐹 “ {𝑃}))
18		fconst3 7214	. 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ⊆ wss 3948  {csn 4628  ∪ cuni 4908  ^◡ccnv 5675   “ cima 5679   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7408  Clsdccld 22519  clsccl 22521   Cn ccn 22727  Frect1 22810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-top 22395  df-topon 22412  df-cld 22522  df-cls 22524  df-cn 22730  df-t1 22817

This theorem is referenced by:  ipasslem8  30085