Theorem cnmpt2res 15321

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 4877	. . . . 5 |- ( ( Y C_ X /\ W C_ Z ) -> ( Y X. W ) C_ ( X X. Z ) )
5	2, 3, 4	syl2anc 415	. . . 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 15286	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ M e. (TopOn ` Z ) ) -> ( J tX M ) e. (TopOn ` ( X X. Z ) ) )
9	6, 7, 8	syl2anc 415	. . . . 5 |- ( ph -> ( J tX M ) e. (TopOn ` ( X X. Z ) ) )
10		toponuni 15039	. . . . 5 |- ( ( J tX M ) e. (TopOn ` ( X X. Z ) ) -> ( X X. Z ) = U. ( J tX M ) )
11	9, 10	syl 14	. . . 4 |- ( ph -> ( X X. Z ) = U. ( J tX M ) )
12	5, 11	sseqtrd 3286	. . 3 |- ( ph -> ( Y X. W ) C_ U. ( J tX M ) )
13		eqid 2238	. . . 4 |- U. ( J tX M ) = U. ( J tX M )
14	13	cnrest 15259	. . 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 415	. 2 |- ( ph -> ( ( x e. X , y e. Z |-> A ) |` ( Y X. W ) ) e. ( ( ( J tX M ) ↾t ( Y X. W ) ) Cn L ) )
16		resmpo 6176	. . 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 415	. 2 |- ( ph -> ( ( x e. X , y e. Z |-> A ) |` ( Y X. W ) ) = ( x e. Y , y e. W |-> A ) )
18		topontop 15038	. . . . . 6 |- ( J e. (TopOn ` X ) -> J e. Top )
19	6, 18	syl 14	. . . . 5 |- ( ph -> J e. Top )
20		topontop 15038	. . . . . 6 |- ( M e. (TopOn ` Z ) -> M e. Top )
21	7, 20	syl 14	. . . . 5 |- ( ph -> M e. Top )
22		toponmax 15049	. . . . . . 7 |- ( J e. (TopOn ` X ) -> X e. J )
23	6, 22	syl 14	. . . . . 6 |- ( ph -> X e. J )
24	23, 2	ssexd 4268	. . . . 5 |- ( ph -> Y e. _V )
25		toponmax 15049	. . . . . . 7 |- ( M e. (TopOn ` Z ) -> Z e. M )
26	7, 25	syl 14	. . . . . 6 |- ( ph -> Z e. M )
27	26, 3	ssexd 4268	. . . . 5 |- ( ph -> W e. _V )
28		txrest 15300	. . . . 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 1279	. . . 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 6087	. . . 4 |- ( K tX N ) = ( ( J ↾t Y ) tX ( M ↾t W ) )
33	29, 32	eqtr4di 2289	. . 3 |- ( ph -> ( ( J tX M ) ↾t ( Y X. W ) ) = ( K tX N ) )
34	33	oveq1d 6090	. 2 |- ( ph -> ( ( ( J tX M ) ↾t ( Y X. W ) ) Cn L ) = ( ( K tX N ) Cn L ) )
35	15, 17, 34	3eltr3d 2321	1 |- ( ph -> ( x e. Y , y e. W |-> A ) e. ( ( K tX N ) Cn L ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   U.cuni 3930   X. cxp 4767   |` cres 4771   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   ↾_t crest 13570   Topctop 15021  TopOnctopon 15034   Cn ccn 15209   tX ctx 15276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-rest 13572  df-topgen 13591  df-top 15022  df-topon 15035  df-bases 15067  df-cn 15212  df-tx 15277

This theorem is referenced by: (None)