Theorem restopnb 15205

Description: If B is an open subset of the subspace base set A, then any subset of B is open iff it is open in A. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
restopnb	|- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. J <-> C e. ( J ↾t A ) ) )

Proof of Theorem restopnb

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr3 1036	. . . . . . 7 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C C_ B )
2		simpr2 1035	. . . . . . 7 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> B C_ A )
3	1, 2	sstrd 3258	. . . . . 6 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C C_ A )
4		df-ss 3233	. . . . . 6 |- ( C C_ A <-> ( C i^i A ) = C )
5	3, 4	sylib 122	. . . . 5 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C i^i A ) = C )
6	5	eqcomd 2244	. . . 4 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C = ( C i^i A ) )
7		ineq1 3425	. . . . . 6 |- ( v = C -> ( v i^i A ) = ( C i^i A ) )
8	7	rspceeqv 2948	. . . . 5 |- ( ( C e. J /\ C = ( C i^i A ) ) -> E. v e. J C = ( v i^i A ) )
9	8	expcom 116	. . . 4 |- ( C = ( C i^i A ) -> ( C e. J -> E. v e. J C = ( v i^i A ) ) )
10	6, 9	syl 14	. . 3 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. J -> E. v e. J C = ( v i^i A ) ) )
11		inass 3441	. . . . . 6 |- ( ( v i^i A ) i^i B ) = ( v i^i ( A i^i B ) )
12		simprr 537	. . . . . . . 8 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> C = ( v i^i A ) )
13	12	ineq1d 3431	. . . . . . 7 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> ( C i^i B ) = ( ( v i^i A ) i^i B ) )
14		simplr3 1072	. . . . . . . . 9 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> C C_ B )
15		df-ss 3233	. . . . . . . . 9 |- ( C C_ B <-> ( C i^i B ) = C )
16	14, 15	sylib 122	. . . . . . . 8 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> ( C i^i B ) = C )
17	16	adantrr 483	. . . . . . 7 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> ( C i^i B ) = C )
18	13, 17	eqtr3d 2273	. . . . . 6 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> ( ( v i^i A ) i^i B ) = C )
19		simplr2 1071	. . . . . . . . 9 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> B C_ A )
20		sseqin2 3450	. . . . . . . . 9 |- ( B C_ A <-> ( A i^i B ) = B )
21	19, 20	sylib 122	. . . . . . . 8 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> ( A i^i B ) = B )
22	21	ineq2d 3432	. . . . . . 7 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> ( v i^i ( A i^i B ) ) = ( v i^i B ) )
23	22	adantrr 483	. . . . . 6 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> ( v i^i ( A i^i B ) ) = ( v i^i B ) )
24	11, 18, 23	3eqtr3a 2295	. . . . 5 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> C = ( v i^i B ) )
25		simplll 539	. . . . . 6 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> J e. Top )
26		simprl 535	. . . . . 6 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> v e. J )
27		simplr1 1070	. . . . . 6 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> B e. J )
28		inopn 15027	. . . . . 6 |- ( ( J e. Top /\ v e. J /\ B e. J ) -> ( v i^i B ) e. J )
29	25, 26, 27, 28	syl3anc 1278	. . . . 5 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> ( v i^i B ) e. J )
30	24, 29	eqeltrd 2315	. . . 4 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> C e. J )
31	30	rexlimdvaa 2669	. . 3 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( E. v e. J C = ( v i^i A ) -> C e. J ) )
32	10, 31	impbid 129	. 2 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. J <-> E. v e. J C = ( v i^i A ) ) )
33		elrest 13577	. . 3 |- ( ( J e. Top /\ A e. V ) -> ( C e. ( J ↾t A ) <-> E. v e. J C = ( v i^i A ) ) )
34	33	adantr 276	. 2 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. ( J ↾t A ) <-> E. v e. J C = ( v i^i A ) ) )
35	32, 34	bitr4d 191	1 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. J <-> C e. ( J ↾t A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   i^i cin 3219   C_ wss 3220  (class class class)co 6075   ↾_t crest 13570   Topctop 15021

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-rest 13572  df-top 15022

This theorem is referenced by:  restopn2  15207