Theorem ressid2 12477

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 993	. . . . 5 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> W e. X )
3	2	elexd 2743	. . . 4 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> W e. _V )
4		simp3 994	. . . . 5 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> A e. Y )
5	4	elexd 2743	. . . 4 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> A e. _V )
6		simp1 992	. . . . . 6 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> B C_ A )
7	6	iftrued 3533	. . . . 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 2247	. . . 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 5500	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = ( Base ` W ) )
11		ressbas.b	. . . . . . . 8 |- B = ( Base ` W )
12	10, 11	eqtr4di 2221	. . . . . . 7 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = B )
13		simpr 109	. . . . . . 7 |- ( ( w = W /\ a = A ) -> a = A )
14	12, 13	sseq12d 3178	. . . . . 6 |- ( ( w = W /\ a = A ) -> ( ( Base ` w ) C_ a <-> B C_ A ) )
15	13, 12	ineq12d 3329	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( a i^i ( Base ` w ) ) = ( A i^i B ) )
16	15	opeq2d 3772	. . . . . . 7 |- ( ( w = W /\ a = A ) -> <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( A i^i B ) >. )
17	9, 16	oveq12d 5871	. . . . . 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 3552	. . . . 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 12424	. . . . 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 5982	. . . 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 1233	. . 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 2203	. 2 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> ( W ↾s A ) = W )
23	1, 22	eqtrid 2215	1 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> R = W )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   _Vcvv 2730   i^i cin 3120   C_ wss 3121   ifcif 3526   <.cop 3586   ` cfv 5198  (class class class)co 5853   ndxcnx 12413   sSet csts 12414   Basecbs 12416   ↾_s cress 12417

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-ress 12424

This theorem is referenced by:  ressid  12479