Theorem restin 14136

Description: When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)

Hypothesis

Ref	Expression
restin.1	|- X = U. J

Assertion

Ref	Expression
restin	|- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ( J ↾t ( A i^i X ) ) )

Proof of Theorem restin

Step	Hyp	Ref	Expression
1		restin.1	. . . . 5 |- X = U. J
2		uniexg 4457	. . . . 5 |- ( J e. V -> U. J e. _V )
3	1, 2	eqeltrid 2276	. . . 4 |- ( J e. V -> X e. _V )
4	3	adantr 276	. . 3 |- ( ( J e. V /\ A e. W ) -> X e. _V )
5		restco 14134	. . . 4 |- ( ( J e. V /\ X e. _V /\ A e. W ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t ( X i^i A ) ) )
6	5	3com23 1211	. . 3 |- ( ( J e. V /\ A e. W /\ X e. _V ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t ( X i^i A ) ) )
7	4, 6	mpd3an3 1349	. 2 |- ( ( J e. V /\ A e. W ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t ( X i^i A ) ) )
8	1	restid 12755	. . . 4 |- ( J e. V -> ( J ↾t X ) = J )
9	8	adantr 276	. . 3 |- ( ( J e. V /\ A e. W ) -> ( J ↾t X ) = J )
10	9	oveq1d 5911	. 2 |- ( ( J e. V /\ A e. W ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t A ) )
11		incom 3342	. . . 4 |- ( X i^i A ) = ( A i^i X )
12	11	oveq2i 5907	. . 3 |- ( J ↾t ( X i^i A ) ) = ( J ↾t ( A i^i X ) )
13	12	a1i 9	. 2 |- ( ( J e. V /\ A e. W ) -> ( J ↾t ( X i^i A ) ) = ( J ↾t ( A i^i X ) ) )
14	7, 10, 13	3eqtr3d 2230	1 |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ( J ↾t ( A i^i X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   _Vcvv 2752   i^i cin 3143   U.cuni 3824  (class class class)co 5896   ↾_t crest 12744

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-rest 12746

This theorem is referenced by:  restuni2  14137  cnrest2r  14197