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Theorem dnsconst 17442
 Description: If a continuous mapping to a T1 space is constant on a dense subset, it is constant on the entire space. Note that means " is dense in " and means " is constant on " (see funconstss 5848). (Contributed by NM, 15-Mar-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
dnsconst.1
dnsconst.2
Assertion
Ref Expression
dnsconst

Proof of Theorem dnsconst
StepHypRef Expression
1 simplr 732 . . 3
2 dnsconst.1 . . . 4
3 dnsconst.2 . . . 4
42, 3cnf 17310 . . 3
5 ffn 5591 . . 3
61, 4, 53syl 19 . 2
7 simpr3 965 . . 3
8 simpll 731 . . . . . 6
9 simpr1 963 . . . . . 6
103t1sncld 17390 . . . . . 6
118, 9, 10syl2anc 643 . . . . 5
12 cnclima 17332 . . . . 5
131, 11, 12syl2anc 643 . . . 4
14 simpr2 964 . . . 4
152clsss2 17136 . . . 4
1613, 14, 15syl2anc 643 . . 3
177, 16eqsstr3d 3383 . 2
18 fconst3 5955 . 2
196, 17, 18sylanbrc 646 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725   wss 3320  csn 3814  cuni 4015  ccnv 4877  cima 4881   wfn 5449  wf 5450  cfv 5454  (class class class)co 6081  ccld 17080  ccl 17082   ccn 17288  ct1 17371 This theorem is referenced by:  ipasslem8  22338 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-top 16963  df-topon 16966  df-cld 17083  df-cls 17085  df-cn 17291  df-t1 17378
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