Theorem restin 15277

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 4585	. . . . 5 |- ( J e. V -> U. J e. _V )
3	1, 2	eqeltrid 2325	. . . 4 |- ( J e. V -> X e. _V )
4	3	adantr 276	. . 3 |- ( ( J e. V /\ A e. W ) -> X e. _V )
5		restco 15275	. . . 4 |- ( ( J e. V /\ X e. _V /\ A e. W ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t ( X i^i A ) ) )
6	5	3com23 1240	. . 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 1379	. 2 |- ( ( J e. V /\ A e. W ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t ( X i^i A ) ) )
8	1	restid 13604	. . . 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 6100	. 2 |- ( ( J e. V /\ A e. W ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t A ) )
11		incom 3421	. . . 4 |- ( X i^i A ) = ( A i^i X )
12	11	oveq2i 6096	. . 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 2279	1 |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ( J ↾t ( A i^i X ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   U.cuni 3935  (class class class)co 6085   ↾_t crest 13593

This proof depends on 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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-rest 13595

This theorem is used by:  restuni2  15278  cnrest2r  15338