Theorem ressid2 12018

Description: General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 26-Jan-2023.)

Hypotheses

Ref	Expression
ressbas.r	|- R = ( W ↾s A )
ressbas.b	|- B = ( Base ` W )

Assertion

Ref	Expression
ressid2	|- ( ( B C_ A /\ W e. X /\ A e. Y ) -> R = W )

Proof of Theorem ressid2

Dummy variables w a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ressbas.r	. 2 |- R = ( W ↾s A )
2		simp2 982	. . . . 5 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> W e. X )
3	2	elexd 2699	. . . 4 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> W e. _V )
4		simp3 983	. . . . 5 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> A e. Y )
5	4	elexd 2699	. . . 4 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> A e. _V )
6		simp1 981	. . . . . 6 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> B C_ A )
7	6	iftrued 3481	. . . . 5 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) = W )
8	7, 3	eqeltrd 2216	. . . 4 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) e. _V )
9		simpl 108	. . . . . . . . 9 |- ( ( w = W /\ a = A ) -> w = W )
10	9	fveq2d 5425	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = ( Base ` W ) )
11		ressbas.b	. . . . . . . 8 |- B = ( Base ` W )
12	10, 11	syl6eqr 2190	. . . . . . 7 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = B )
13		simpr 109	. . . . . . 7 |- ( ( w = W /\ a = A ) -> a = A )
14	12, 13	sseq12d 3128	. . . . . 6 |- ( ( w = W /\ a = A ) -> ( ( Base ` w ) C_ a <-> B C_ A ) )
15	13, 12	ineq12d 3278	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( a i^i ( Base ` w ) ) = ( A i^i B ) )
16	15	opeq2d 3712	. . . . . . 7 |- ( ( w = W /\ a = A ) -> <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( A i^i B ) >. )
17	9, 16	oveq12d 5792	. . . . . 6 |- ( ( w = W /\ a = A ) -> ( w sSet <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) )
18	14, 9, 17	ifbieq12d 3498	. . . . 5 |- ( ( w = W /\ a = A ) -> if ( ( Base ` w ) C_ a , w , ( w sSet <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. ) ) = if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) )
19		df-ress 11967	. . . . 5 |- ↾s = ( w e. _V , a e. _V |-> if ( ( Base ` w ) C_ a , w , ( w sSet <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. ) ) )
20	18, 19	ovmpoga 5900	. . . 4 |- ( ( W e. _V /\ A e. _V /\ if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) e. _V ) -> ( W ↾s A ) = if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) )
21	3, 5, 8, 20	syl3anc 1216	. . 3 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> ( W ↾s A ) = if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) )
22	21, 7	eqtrd 2172	. 2 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> ( W ↾s A ) = W )
23	1, 22	syl5eq 2184	1 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> R = W )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   _Vcvv 2686   i^i cin 3070   C_ wss 3071   ifcif 3474   <.cop 3530   ` cfv 5123  (class class class)co 5774   ndxcnx 11956   sSet csts 11957   Basecbs 11959   ↾_s cress 11960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-ress 11967

This theorem is referenced by:  ressid  12020