Theorem dnsconst 21986

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 6826). (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 21854	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌)
5		ffn 6514	. . 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 1192	. . 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 1190	. . . . . 6 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝑃 ∈ 𝑌)
10	3	t1sncld 21934	. . . . . 6 ⊢ ((𝐾 ∈ Fre ∧ 𝑃 ∈ 𝑌) → {𝑃} ∈ (Clsd‘𝐾))
11	8, 9, 10	syl2anc 586	. . . . 5 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → {𝑃} ∈ (Clsd‘𝐾))
12		cnclima 21876	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ {𝑃} ∈ (Clsd‘𝐾)) → (◡𝐹 “ {𝑃}) ∈ (Clsd‘𝐽))
13	1, 11, 12	syl2anc 586	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (◡𝐹 “ {𝑃}) ∈ (Clsd‘𝐽))
14		simpr2 1191	. . . 4 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ⊆ (◡𝐹 “ {𝑃}))
15	2	clsss2 21680	. . . 4 ⊢ (((◡𝐹 “ {𝑃}) ∈ (Clsd‘𝐽) ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃})) → ((cls‘𝐽)‘𝐴) ⊆ (◡𝐹 “ {𝑃}))
16	13, 14, 15	syl2anc 586	. . 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 4006	. 2 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝑋 ⊆ (◡𝐹 “ {𝑃}))
18		fconst3 6976	. 2 ⊢ (𝐹:𝑋⟶{𝑃} ↔ (𝐹 Fn 𝑋 ∧ 𝑋 ⊆ (◡𝐹 “ {𝑃})))
19	6, 17, 18	sylanbrc 585	1 ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹:𝑋⟶{𝑃})

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ⊆ wss 3936  {csn 4567  ∪ cuni 4838  ^◡ccnv 5554   “ cima 5558   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  Clsdccld 21624  clsccl 21626   Cn ccn 21832  Frect1 21915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8408  df-top 21502  df-topon 21519  df-cld 21627  df-cls 21629  df-cn 21835  df-t1 21922

This theorem is referenced by:  ipasslem8  28614