Theorem restin 14412

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 4474	. . . . 5 |- ( J e. V -> U. J e. _V )
3	1, 2	eqeltrid 2283	. . . 4 |- ( J e. V -> X e. _V )
4	3	adantr 276	. . 3 |- ( ( J e. V /\ A e. W ) -> X e. _V )
5		restco 14410	. . . 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 12921	. . . 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 5937	. 2 |- ( ( J e. V /\ A e. W ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t A ) )
11		incom 3355	. . . 4 |- ( X i^i A ) = ( A i^i X )
12	11	oveq2i 5933	. . 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 2237	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 2167   _Vcvv 2763   i^i cin 3156   U.cuni 3839  (class class class)co 5922   ↾_t crest 12910

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-rest 12912

This theorem is referenced by:  restuni2  14413  cnrest2r  14473