Theorem cnmpt2res 21480

Description: The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)

Hypotheses

Ref	Expression
cnmpt1res.2	|- K = ( J ↾t Y )
cnmpt1res.3	|- ( ph -> J e. (TopOn ` X ) )
cnmpt1res.5	|- ( ph -> Y C_ X )
cnmpt2res.7	|- N = ( M ↾t W )
cnmpt2res.8	|- ( ph -> M e. (TopOn ` Z ) )
cnmpt2res.9	|- ( ph -> W C_ Z )
cnmpt2res.10	|- ( ph -> ( x e. X , y e. Z |-> A ) e. ( ( J tX M ) Cn L ) )

Assertion

Ref	Expression
cnmpt2res	|- ( ph -> ( x e. Y , y e. W |-> A ) e. ( ( K tX N ) Cn L ) )

Distinct variable groups:   x, y, W   x, X, y   x, Y, y   x, Z, y

Allowed substitution hints:   ph( x, y)   A( x, y)   J( x, y)   K( x, y)   L( x, y)   M( x, y)   N( x, y)

Proof of Theorem cnmpt2res

Step	Hyp	Ref	Expression
1		cnmpt2res.10	. . 3 |- ( ph -> ( x e. X , y e. Z |-> A ) e. ( ( J tX M ) Cn L ) )
2		cnmpt1res.5	. . . . 5 |- ( ph -> Y C_ X )
3		cnmpt2res.9	. . . . 5 |- ( ph -> W C_ Z )
4		xpss12 5225	. . . . 5 |- ( ( Y C_ X /\ W C_ Z ) -> ( Y X. W ) C_ ( X X. Z ) )
5	2, 3, 4	syl2anc 693	. . . 4 |- ( ph -> ( Y X. W ) C_ ( X X. Z ) )
6		cnmpt1res.3	. . . . . 6 |- ( ph -> J e. (TopOn ` X ) )
7		cnmpt2res.8	. . . . . 6 |- ( ph -> M e. (TopOn ` Z ) )
8		txtopon 21394	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ M e. (TopOn ` Z ) ) -> ( J tX M ) e. (TopOn ` ( X X. Z ) ) )
9	6, 7, 8	syl2anc 693	. . . . 5 |- ( ph -> ( J tX M ) e. (TopOn ` ( X X. Z ) ) )
10		toponuni 20719	. . . . 5 |- ( ( J tX M ) e. (TopOn ` ( X X. Z ) ) -> ( X X. Z ) = U. ( J tX M ) )
11	9, 10	syl 17	. . . 4 |- ( ph -> ( X X. Z ) = U. ( J tX M ) )
12	5, 11	sseqtrd 3641	. . 3 |- ( ph -> ( Y X. W ) C_ U. ( J tX M ) )
13		eqid 2622	. . . 4 |- U. ( J tX M ) = U. ( J tX M )
14	13	cnrest 21089	. . 3 |- ( ( ( x e. X , y e. Z |-> A ) e. ( ( J tX M ) Cn L ) /\ ( Y X. W ) C_ U. ( J tX M ) ) -> ( ( x e. X , y e. Z |-> A ) |` ( Y X. W ) ) e. ( ( ( J tX M ) ↾t ( Y X. W ) ) Cn L ) )
15	1, 12, 14	syl2anc 693	. 2 |- ( ph -> ( ( x e. X , y e. Z |-> A ) |` ( Y X. W ) ) e. ( ( ( J tX M ) ↾t ( Y X. W ) ) Cn L ) )
16		resmpt2 6758	. . 3 |- ( ( Y C_ X /\ W C_ Z ) -> ( ( x e. X , y e. Z |-> A ) |` ( Y X. W ) ) = ( x e. Y , y e. W |-> A ) )
17	2, 3, 16	syl2anc 693	. 2 |- ( ph -> ( ( x e. X , y e. Z |-> A ) |` ( Y X. W ) ) = ( x e. Y , y e. W |-> A ) )
18		topontop 20718	. . . . . 6 |- ( J e. (TopOn ` X ) -> J e. Top )
19	6, 18	syl 17	. . . . 5 |- ( ph -> J e. Top )
20		topontop 20718	. . . . . 6 |- ( M e. (TopOn ` Z ) -> M e. Top )
21	7, 20	syl 17	. . . . 5 |- ( ph -> M e. Top )
22		toponmax 20730	. . . . . . 7 |- ( J e. (TopOn ` X ) -> X e. J )
23	6, 22	syl 17	. . . . . 6 |- ( ph -> X e. J )
24	23, 2	ssexd 4805	. . . . 5 |- ( ph -> Y e. _V )
25		toponmax 20730	. . . . . . 7 |- ( M e. (TopOn ` Z ) -> Z e. M )
26	7, 25	syl 17	. . . . . 6 |- ( ph -> Z e. M )
27	26, 3	ssexd 4805	. . . . 5 |- ( ph -> W e. _V )
28		txrest 21434	. . . . 5 |- ( ( ( J e. Top /\ M e. Top ) /\ ( Y e. _V /\ W e. _V ) ) -> ( ( J tX M ) ↾t ( Y X. W ) ) = ( ( J ↾t Y ) tX ( M ↾t W ) ) )
29	19, 21, 24, 27, 28	syl22anc 1327	. . . 4 |- ( ph -> ( ( J tX M ) ↾t ( Y X. W ) ) = ( ( J ↾t Y ) tX ( M ↾t W ) ) )
30		cnmpt1res.2	. . . . 5 |- K = ( J ↾t Y )
31		cnmpt2res.7	. . . . 5 |- N = ( M ↾t W )
32	30, 31	oveq12i 6662	. . . 4 |- ( K tX N ) = ( ( J ↾t Y ) tX ( M ↾t W ) )
33	29, 32	syl6eqr 2674	. . 3 |- ( ph -> ( ( J tX M ) ↾t ( Y X. W ) ) = ( K tX N ) )
34	33	oveq1d 6665	. 2 |- ( ph -> ( ( ( J tX M ) ↾t ( Y X. W ) ) Cn L ) = ( ( K tX N ) Cn L ) )
35	15, 17, 34	3eltr3d 2715	1 |- ( ph -> ( x e. Y , y e. W |-> A ) e. ( ( K tX N ) Cn L ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   U.cuni 4436   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363

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-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-cn 21031  df-tx 21365

This theorem is referenced by:  symgtgp  21905  submtmd  21908  iimulcn  22737  cxpcn2  24487  cxpcn3  24489  cvxsconn  31225  cvmlift2lem6  31290  cvmlift2lem12  31296