Theorem paste 21098

Description: Pasting lemma. If A and B are closed sets in X with A u. B = X, then any function whose restrictions to A and B are continuous is continuous on all of X. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Hypotheses

Ref	Expression
paste.1	|- X = U. J
paste.2	|- Y = U. K
paste.4	|- ( ph -> A e. ( Clsd ` J ) )
paste.5	|- ( ph -> B e. ( Clsd ` J ) )
paste.6	|- ( ph -> ( A u. B ) = X )
paste.7	|- ( ph -> F : X --> Y )
paste.8	|- ( ph -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
paste.9	|- ( ph -> ( F |` B ) e. ( ( J ↾t B ) Cn K ) )

Assertion

Ref	Expression
paste	|- ( ph -> F e. ( J Cn K ) )

Proof of Theorem paste

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		paste.7	. 2 |- ( ph -> F : X --> Y )
2		paste.6	. . . . . . 7 |- ( ph -> ( A u. B ) = X )
3	2	ineq2d 3814	. . . . . 6 |- ( ph -> ( ( `' F " y ) i^i ( A u. B ) ) = ( ( `' F " y ) i^i X ) )
4		ffun 6048	. . . . . . . . 9 |- ( F : X --> Y -> Fun F )
5	1, 4	syl 17	. . . . . . . 8 |- ( ph -> Fun F )
6		respreima 6344	. . . . . . . . 9 |- ( Fun F -> ( `' ( F |` A ) " y ) = ( ( `' F " y ) i^i A ) )
7		respreima 6344	. . . . . . . . 9 |- ( Fun F -> ( `' ( F |` B ) " y ) = ( ( `' F " y ) i^i B ) )
8	6, 7	uneq12d 3768	. . . . . . . 8 |- ( Fun F -> ( ( `' ( F |` A ) " y ) u. ( `' ( F |` B ) " y ) ) = ( ( ( `' F " y ) i^i A ) u. ( ( `' F " y ) i^i B ) ) )
9	5, 8	syl 17	. . . . . . 7 |- ( ph -> ( ( `' ( F |` A ) " y ) u. ( `' ( F |` B ) " y ) ) = ( ( ( `' F " y ) i^i A ) u. ( ( `' F " y ) i^i B ) ) )
10		indi 3873	. . . . . . 7 |- ( ( `' F " y ) i^i ( A u. B ) ) = ( ( ( `' F " y ) i^i A ) u. ( ( `' F " y ) i^i B ) )
11	9, 10	syl6reqr 2675	. . . . . 6 |- ( ph -> ( ( `' F " y ) i^i ( A u. B ) ) = ( ( `' ( F |` A ) " y ) u. ( `' ( F |` B ) " y ) ) )
12		imassrn 5477	. . . . . . . . 9 |- ( `' F " y ) C_ ran `' F
13		dfdm4 5316	. . . . . . . . . 10 |- dom F = ran `' F
14		fdm 6051	. . . . . . . . . 10 |- ( F : X --> Y -> dom F = X )
15	13, 14	syl5eqr 2670	. . . . . . . . 9 |- ( F : X --> Y -> ran `' F = X )
16	12, 15	syl5sseq 3653	. . . . . . . 8 |- ( F : X --> Y -> ( `' F " y ) C_ X )
17	1, 16	syl 17	. . . . . . 7 |- ( ph -> ( `' F " y ) C_ X )
18		df-ss 3588	. . . . . . 7 |- ( ( `' F " y ) C_ X <-> ( ( `' F " y ) i^i X ) = ( `' F " y ) )
19	17, 18	sylib 208	. . . . . 6 |- ( ph -> ( ( `' F " y ) i^i X ) = ( `' F " y ) )
20	3, 11, 19	3eqtr3rd 2665	. . . . 5 |- ( ph -> ( `' F " y ) = ( ( `' ( F |` A ) " y ) u. ( `' ( F |` B ) " y ) ) )
21	20	adantr 481	. . . 4 |- ( ( ph /\ y e. ( Clsd ` K ) ) -> ( `' F " y ) = ( ( `' ( F |` A ) " y ) u. ( `' ( F |` B ) " y ) ) )
22		paste.4	. . . . . . 7 |- ( ph -> A e. ( Clsd ` J ) )
23	22	adantr 481	. . . . . 6 |- ( ( ph /\ y e. ( Clsd ` K ) ) -> A e. ( Clsd ` J ) )
24		paste.8	. . . . . . 7 |- ( ph -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
25		cnclima 21072	. . . . . . 7 |- ( ( ( F |` A ) e. ( ( J ↾t A ) Cn K ) /\ y e. ( Clsd ` K ) ) -> ( `' ( F |` A ) " y ) e. ( Clsd ` ( J ↾t A ) ) )
26	24, 25	sylan 488	. . . . . 6 |- ( ( ph /\ y e. ( Clsd ` K ) ) -> ( `' ( F |` A ) " y ) e. ( Clsd ` ( J ↾t A ) ) )
27		restcldr 20978	. . . . . 6 |- ( ( A e. ( Clsd ` J ) /\ ( `' ( F |` A ) " y ) e. ( Clsd ` ( J ↾t A ) ) ) -> ( `' ( F |` A ) " y ) e. ( Clsd ` J ) )
28	23, 26, 27	syl2anc 693	. . . . 5 |- ( ( ph /\ y e. ( Clsd ` K ) ) -> ( `' ( F |` A ) " y ) e. ( Clsd ` J ) )
29		paste.5	. . . . . . 7 |- ( ph -> B e. ( Clsd ` J ) )
30	29	adantr 481	. . . . . 6 |- ( ( ph /\ y e. ( Clsd ` K ) ) -> B e. ( Clsd ` J ) )
31		paste.9	. . . . . . 7 |- ( ph -> ( F |` B ) e. ( ( J ↾t B ) Cn K ) )
32		cnclima 21072	. . . . . . 7 |- ( ( ( F |` B ) e. ( ( J ↾t B ) Cn K ) /\ y e. ( Clsd ` K ) ) -> ( `' ( F |` B ) " y ) e. ( Clsd ` ( J ↾t B ) ) )
33	31, 32	sylan 488	. . . . . 6 |- ( ( ph /\ y e. ( Clsd ` K ) ) -> ( `' ( F |` B ) " y ) e. ( Clsd ` ( J ↾t B ) ) )
34		restcldr 20978	. . . . . 6 |- ( ( B e. ( Clsd ` J ) /\ ( `' ( F |` B ) " y ) e. ( Clsd ` ( J ↾t B ) ) ) -> ( `' ( F |` B ) " y ) e. ( Clsd ` J ) )
35	30, 33, 34	syl2anc 693	. . . . 5 |- ( ( ph /\ y e. ( Clsd ` K ) ) -> ( `' ( F |` B ) " y ) e. ( Clsd ` J ) )
36		uncld 20845	. . . . 5 |- ( ( ( `' ( F |` A ) " y ) e. ( Clsd ` J ) /\ ( `' ( F |` B ) " y ) e. ( Clsd ` J ) ) -> ( ( `' ( F |` A ) " y ) u. ( `' ( F |` B ) " y ) ) e. ( Clsd ` J ) )
37	28, 35, 36	syl2anc 693	. . . 4 |- ( ( ph /\ y e. ( Clsd ` K ) ) -> ( ( `' ( F |` A ) " y ) u. ( `' ( F |` B ) " y ) ) e. ( Clsd ` J ) )
38	21, 37	eqeltrd 2701	. . 3 |- ( ( ph /\ y e. ( Clsd ` K ) ) -> ( `' F " y ) e. ( Clsd ` J ) )
39	38	ralrimiva 2966	. 2 |- ( ph -> A. y e. ( Clsd ` K ) ( `' F " y ) e. ( Clsd ` J ) )
40		cldrcl 20830	. . . 4 |- ( A e. ( Clsd ` J ) -> J e. Top )
41	22, 40	syl 17	. . 3 |- ( ph -> J e. Top )
42		cntop2 21045	. . . 4 |- ( ( F |` A ) e. ( ( J ↾t A ) Cn K ) -> K e. Top )
43	24, 42	syl 17	. . 3 |- ( ph -> K e. Top )
44		paste.1	. . . . 5 |- X = U. J
45	44	toptopon 20722	. . . 4 |- ( J e. Top <-> J e. (TopOn ` X ) )
46		paste.2	. . . . 5 |- Y = U. K
47	46	toptopon 20722	. . . 4 |- ( K e. Top <-> K e. (TopOn ` Y ) )
48		iscncl 21073	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. y e. ( Clsd ` K ) ( `' F " y ) e. ( Clsd ` J ) ) ) )
49	45, 47, 48	syl2anb 496	. . 3 |- ( ( J e. Top /\ K e. Top ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. y e. ( Clsd ` K ) ( `' F " y ) e. ( Clsd ` J ) ) ) )
50	41, 43, 49	syl2anc 693	. 2 |- ( ph -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. y e. ( Clsd ` K ) ( `' F " y ) e. ( Clsd ` J ) ) ) )
51	1, 39, 50	mpbir2and 957	1 |- ( ph -> F e. ( J Cn K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   u. cun 3572   i^i cin 3573   C_ wss 3574   U.cuni 4436   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   Clsdccld 20820   Cn ccn 21028

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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-cn 21031

This theorem is referenced by:  cnmpt2pc  22727  cvmliftlem10  31276