Theorem restopnb 14768

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 1008	. . . . . . 7 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C C_ B )
2		simpr2 1007	. . . . . . 7 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> B C_ A )
3	1, 2	sstrd 3211	. . . . . 6 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C C_ A )
4		df-ss 3187	. . . . . 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 2213	. . . 4 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C = ( C i^i A ) )
7		ineq1 3375	. . . . . 6 |- ( v = C -> ( v i^i A ) = ( C i^i A ) )
8	7	rspceeqv 2902	. . . . 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 3391	. . . . . 6 |- ( ( v i^i A ) i^i B ) = ( v i^i ( A i^i B ) )
12		simprr 531	. . . . . . . 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 3381	. . . . . . 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 1044	. . . . . . . . 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 3187	. . . . . . . . 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 479	. . . . . . 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 2242	. . . . . 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 1043	. . . . . . . . 9 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> B C_ A )
20		sseqin2 3400	. . . . . . . . 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 3382	. . . . . . 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 479	. . . . . 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 2264	. . . . 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 533	. . . . . 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 529	. . . . . 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 1042	. . . . . 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 14590	. . . . . 6 |- ( ( J e. Top /\ v e. J /\ B e. J ) -> ( v i^i B ) e. J )
29	25, 26, 27, 28	syl3anc 1250	. . . . 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 2284	. . . 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 2626	. . 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 13193	. . 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 981   = wceq 1373   e. wcel 2178   E.wrex 2487   i^i cin 3173   C_ wss 3174  (class class class)co 5967   ↾_t crest 13186   Topctop 14584

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-rest 13188  df-top 14585

This theorem is referenced by:  restopn2  14770