Theorem ressid2 12319

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 983	. . . . 5 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> W e. X )
3	2	elexd 2725	. . . 4 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> W e. _V )
4		simp3 984	. . . . 5 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> A e. Y )
5	4	elexd 2725	. . . 4 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> A e. _V )
6		simp1 982	. . . . . 6 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> B C_ A )
7	6	iftrued 3513	. . . . 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 2234	. . . 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 5474	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = ( Base ` W ) )
11		ressbas.b	. . . . . . . 8 |- B = ( Base ` W )
12	10, 11	eqtr4di 2208	. . . . . . 7 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = B )
13		simpr 109	. . . . . . 7 |- ( ( w = W /\ a = A ) -> a = A )
14	12, 13	sseq12d 3159	. . . . . 6 |- ( ( w = W /\ a = A ) -> ( ( Base ` w ) C_ a <-> B C_ A ) )
15	13, 12	ineq12d 3310	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( a i^i ( Base ` w ) ) = ( A i^i B ) )
16	15	opeq2d 3750	. . . . . . 7 |- ( ( w = W /\ a = A ) -> <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( A i^i B ) >. )
17	9, 16	oveq12d 5844	. . . . . 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 3532	. . . . 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 12268	. . . . 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 5952	. . . 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 1220	. . 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 2190	. 2 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> ( W ↾s A ) = W )
23	1, 22	syl5eq 2202	1 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> R = W )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1335   e. wcel 2128   _Vcvv 2712   i^i cin 3101   C_ wss 3102   ifcif 3506   <.cop 3564   ` cfv 5172  (class class class)co 5826   ndxcnx 12257   sSet csts 12258   Basecbs 12260   ↾_s cress 12261

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-setind 4498

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-iota 5137  df-fun 5174  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-ress 12268

This theorem is referenced by:  ressid  12321