Theorem restin 20970

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 6955	. . . . 5 |- ( J e. V -> U. J e. _V )
3	1, 2	syl5eqel 2705	. . . 4 |- ( J e. V -> X e. _V )
4	3	adantr 481	. . 3 |- ( ( J e. V /\ A e. W ) -> X e. _V )
5		restco 20968	. . . 4 |- ( ( J e. V /\ X e. _V /\ A e. W ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t ( X i^i A ) ) )
6	5	3com23 1271	. . 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 1425	. 2 |- ( ( J e. V /\ A e. W ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t ( X i^i A ) ) )
8	1	restid 16094	. . . 4 |- ( J e. V -> ( J ↾t X ) = J )
9	8	adantr 481	. . 3 |- ( ( J e. V /\ A e. W ) -> ( J ↾t X ) = J )
10	9	oveq1d 6665	. 2 |- ( ( J e. V /\ A e. W ) -> ( ( J ↾t X ) ↾t A ) = ( J ↾t A ) )
11		incom 3805	. . . 4 |- ( X i^i A ) = ( A i^i X )
12	11	oveq2i 6661	. . 3 |- ( J ↾t ( X i^i A ) ) = ( J ↾t ( A i^i X ) )
13	12	a1i 11	. 2 |- ( ( J e. V /\ A e. W ) -> ( J ↾t ( X i^i A ) ) = ( J ↾t ( A i^i X ) ) )
14	7, 10, 13	3eqtr3d 2664	1 |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ( J ↾t ( A i^i X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   U.cuni 4436  (class class class)co 6650   ↾_t crest 16081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rest 16083

This theorem is referenced by:  restuni2  20971  cnrest2r  21091  cnrmi  21164  restcnrm  21166  resthauslem  21167  imacmp  21200  fiuncmp  21207  kgeni  21340  ressxms  22330  ptrest  33408  restuni6  39305