Theorem cnmpt2res 13348

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 4727	. . . . 5 |- ( ( Y C_ X /\ W C_ Z ) -> ( Y X. W ) C_ ( X X. Z ) )
5	2, 3, 4	syl2anc 411	. . . 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 13313	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ M e. (TopOn ` Z ) ) -> ( J tX M ) e. (TopOn ` ( X X. Z ) ) )
9	6, 7, 8	syl2anc 411	. . . . 5 |- ( ph -> ( J tX M ) e. (TopOn ` ( X X. Z ) ) )
10		toponuni 13064	. . . . 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 3191	. . 3 |- ( ph -> ( Y X. W ) C_ U. ( J tX M ) )
13		eqid 2175	. . . 4 |- U. ( J tX M ) = U. ( J tX M )
14	13	cnrest 13286	. . 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 411	. 2 |- ( ph -> ( ( x e. X , y e. Z |-> A ) |` ( Y X. W ) ) e. ( ( ( J tX M ) ↾t ( Y X. W ) ) Cn L ) )
16		resmpo 5963	. . 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 411	. 2 |- ( ph -> ( ( x e. X , y e. Z |-> A ) |` ( Y X. W ) ) = ( x e. Y , y e. W |-> A ) )
18		topontop 13063	. . . . . 6 |- ( J e. (TopOn ` X ) -> J e. Top )
19	6, 18	syl 14	. . . . 5 |- ( ph -> J e. Top )
20		topontop 13063	. . . . . 6 |- ( M e. (TopOn ` Z ) -> M e. Top )
21	7, 20	syl 14	. . . . 5 |- ( ph -> M e. Top )
22		toponmax 13074	. . . . . . 7 |- ( J e. (TopOn ` X ) -> X e. J )
23	6, 22	syl 14	. . . . . 6 |- ( ph -> X e. J )
24	23, 2	ssexd 4138	. . . . 5 |- ( ph -> Y e. _V )
25		toponmax 13074	. . . . . . 7 |- ( M e. (TopOn ` Z ) -> Z e. M )
26	7, 25	syl 14	. . . . . 6 |- ( ph -> Z e. M )
27	26, 3	ssexd 4138	. . . . 5 |- ( ph -> W e. _V )
28		txrest 13327	. . . . 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 1239	. . . 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 5877	. . . 4 |- ( K tX N ) = ( ( J ↾t Y ) tX ( M ↾t W ) )
33	29, 32	eqtr4di 2226	. . 3 |- ( ph -> ( ( J tX M ) ↾t ( Y X. W ) ) = ( K tX N ) )
34	33	oveq1d 5880	. 2 |- ( ph -> ( ( ( J tX M ) ↾t ( Y X. W ) ) Cn L ) = ( ( K tX N ) Cn L ) )
35	15, 17, 34	3eltr3d 2258	1 |- ( ph -> ( x e. Y , y e. W |-> A ) e. ( ( K tX N ) Cn L ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2146   _Vcvv 2735   C_ wss 3127   U.cuni 3805   X. cxp 4618   |` cres 4622   ` cfv 5208  (class class class)co 5865   e. cmpo 5867   ↾_t crest 12608   Topctop 13046  TopOnctopon 13059   Cn ccn 13236   tX ctx 13303

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-map 6640  df-rest 12610  df-topgen 12629  df-top 13047  df-topon 13060  df-bases 13092  df-cn 13239  df-tx 13304

This theorem is referenced by: (None)