Theorem restin 14344

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 4470	. . . . 5 |- ( J e. V -> U. J e. _V )
3	1, 2	eqeltrid 2280	. . . 4 |- ( J e. V -> X e. _V )
4	3	adantr 276	. . 3 |- ( ( J e. V /\ A e. W ) -> X e. _V )
5		restco 14342	. . . 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 12861	. . . 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 5933	. 2 |- ( ( J e. V /\ A e. W ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t A ) )
11		incom 3351	. . . 4 |- ( X i^i A ) = ( A i^i X )
12	11	oveq2i 5929	. . 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 2234	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 2164   _Vcvv 2760   i^i cin 3152   U.cuni 3835  (class class class)co 5918   ↾_t crest 12850

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-rest 12852

This theorem is referenced by:  restuni2  14345  cnrest2r  14405