Theorem cnmpt2res 12937

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 4711	. . . . 5 |- ( ( Y C_ X /\ W C_ Z ) -> ( Y X. W ) C_ ( X X. Z ) )
5	2, 3, 4	syl2anc 409	. . . 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 12902	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ M e. (TopOn ` Z ) ) -> ( J tX M ) e. (TopOn ` ( X X. Z ) ) )
9	6, 7, 8	syl2anc 409	. . . . 5 |- ( ph -> ( J tX M ) e. (TopOn ` ( X X. Z ) ) )
10		toponuni 12653	. . . . 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 3180	. . 3 |- ( ph -> ( Y X. W ) C_ U. ( J tX M ) )
13		eqid 2165	. . . 4 |- U. ( J tX M ) = U. ( J tX M )
14	13	cnrest 12875	. . 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 409	. 2 |- ( ph -> ( ( x e. X , y e. Z |-> A ) |` ( Y X. W ) ) e. ( ( ( J tX M ) ↾t ( Y X. W ) ) Cn L ) )
16		resmpo 5940	. . 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 409	. 2 |- ( ph -> ( ( x e. X , y e. Z |-> A ) |` ( Y X. W ) ) = ( x e. Y , y e. W |-> A ) )
18		topontop 12652	. . . . . 6 |- ( J e. (TopOn ` X ) -> J e. Top )
19	6, 18	syl 14	. . . . 5 |- ( ph -> J e. Top )
20		topontop 12652	. . . . . 6 |- ( M e. (TopOn ` Z ) -> M e. Top )
21	7, 20	syl 14	. . . . 5 |- ( ph -> M e. Top )
22		toponmax 12663	. . . . . . 7 |- ( J e. (TopOn ` X ) -> X e. J )
23	6, 22	syl 14	. . . . . 6 |- ( ph -> X e. J )
24	23, 2	ssexd 4122	. . . . 5 |- ( ph -> Y e. _V )
25		toponmax 12663	. . . . . . 7 |- ( M e. (TopOn ` Z ) -> Z e. M )
26	7, 25	syl 14	. . . . . 6 |- ( ph -> Z e. M )
27	26, 3	ssexd 4122	. . . . 5 |- ( ph -> W e. _V )
28		txrest 12916	. . . . 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 1229	. . . 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 5854	. . . 4 |- ( K tX N ) = ( ( J ↾t Y ) tX ( M ↾t W ) )
33	29, 32	eqtr4di 2217	. . 3 |- ( ph -> ( ( J tX M ) ↾t ( Y X. W ) ) = ( K tX N ) )
34	33	oveq1d 5857	. 2 |- ( ph -> ( ( ( J tX M ) ↾t ( Y X. W ) ) Cn L ) = ( ( K tX N ) Cn L ) )
35	15, 17, 34	3eltr3d 2249	1 |- ( ph -> ( x e. Y , y e. W |-> A ) e. ( ( K tX N ) Cn L ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   U.cuni 3789   X. cxp 4602   |` cres 4606   ` cfv 5188  (class class class)co 5842   e. cmpo 5844   ↾_t crest 12556   Topctop 12635  TopOnctopon 12648   Cn ccn 12825   tX ctx 12892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-rest 12558  df-topgen 12577  df-top 12636  df-topon 12649  df-bases 12681  df-cn 12828  df-tx 12893

This theorem is referenced by: (None)