Theorem dnsconst 21182

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 C_ ( `' F " { P } ) means " F is constant on A " (see funconstss 6335). (Contributed by NM, 15-Mar-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Hypotheses

Ref	Expression
dnsconst.1	|- X = U. J
dnsconst.2	|- Y = U. K

Assertion

Ref	Expression
dnsconst	|- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> F : X --> { P } )

Proof of Theorem dnsconst

Step	Hyp	Ref	Expression
1		simplr 792	. . 3 |- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> F e. ( J Cn K ) )
2		dnsconst.1	. . . 4 |- X = U. J
3		dnsconst.2	. . . 4 |- Y = U. K
4	2, 3	cnf 21050	. . 3 |- ( F e. ( J Cn K ) -> F : X --> Y )
5		ffn 6045	. . 3 |- ( F : X --> Y -> F Fn X )
6	1, 4, 5	3syl 18	. 2 |- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> F Fn X )
7		simpr3 1069	. . 3 |- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> ( ( cls ` J ) ` A ) = X )
8		simpll 790	. . . . . 6 |- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> K e. Fre )
9		simpr1 1067	. . . . . 6 |- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> P e. Y )
10	3	t1sncld 21130	. . . . . 6 |- ( ( K e. Fre /\ P e. Y ) -> { P } e. ( Clsd ` K ) )
11	8, 9, 10	syl2anc 693	. . . . 5 |- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> { P } e. ( Clsd ` K ) )
12		cnclima 21072	. . . . 5 |- ( ( F e. ( J Cn K ) /\ { P } e. ( Clsd ` K ) ) -> ( `' F " { P } ) e. ( Clsd ` J ) )
13	1, 11, 12	syl2anc 693	. . . 4 |- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> ( `' F " { P } ) e. ( Clsd ` J ) )
14		simpr2 1068	. . . 4 |- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> A C_ ( `' F " { P } ) )
15	2	clsss2 20876	. . . 4 |- ( ( ( `' F " { P } ) e. ( Clsd ` J ) /\ A C_ ( `' F " { P } ) ) -> ( ( cls ` J ) ` A ) C_ ( `' F " { P } ) )
16	13, 14, 15	syl2anc 693	. . 3 |- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> ( ( cls ` J ) ` A ) C_ ( `' F " { P } ) )
17	7, 16	eqsstr3d 3640	. 2 |- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> X C_ ( `' F " { P } ) )
18		fconst3 6477	. 2 |- ( F : X --> { P } <-> ( F Fn X /\ X C_ ( `' F " { P } ) ) )
19	6, 17, 18	sylanbrc 698	1 |- ( ( ( K e. Fre /\ F e. ( J Cn K ) ) /\ ( P e. Y /\ A C_ ( `' F " { P } ) /\ ( ( cls ` J ) ` A ) = X ) ) -> F : X --> { P } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   U.cuni 4436   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Clsdccld 20820   clsccl 20822   Cn ccn 21028   Frect1 21111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-top 20699  df-topon 20716  df-cld 20823  df-cls 20825  df-cn 21031  df-t1 21118

This theorem is referenced by:  ipasslem8  27692