Theorem ressid2 12454

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 988	. . . . 5 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> W e. X )
3	2	elexd 2739	. . . 4 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> W e. _V )
4		simp3 989	. . . . 5 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> A e. Y )
5	4	elexd 2739	. . . 4 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> A e. _V )
6		simp1 987	. . . . . 6 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> B C_ A )
7	6	iftrued 3527	. . . . 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 2243	. . . 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 5490	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = ( Base ` W ) )
11		ressbas.b	. . . . . . . 8 |- B = ( Base ` W )
12	10, 11	eqtr4di 2217	. . . . . . 7 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = B )
13		simpr 109	. . . . . . 7 |- ( ( w = W /\ a = A ) -> a = A )
14	12, 13	sseq12d 3173	. . . . . 6 |- ( ( w = W /\ a = A ) -> ( ( Base ` w ) C_ a <-> B C_ A ) )
15	13, 12	ineq12d 3324	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( a i^i ( Base ` w ) ) = ( A i^i B ) )
16	15	opeq2d 3765	. . . . . . 7 |- ( ( w = W /\ a = A ) -> <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( A i^i B ) >. )
17	9, 16	oveq12d 5860	. . . . . 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 3546	. . . . 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 12402	. . . . 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 5971	. . . 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 1228	. . 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 2198	. 2 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> ( W ↾s A ) = W )
23	1, 22	syl5eq 2211	1 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> R = W )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   _Vcvv 2726   i^i cin 3115   C_ wss 3116   ifcif 3520   <.cop 3579   ` cfv 5188  (class class class)co 5842   ndxcnx 12391   sSet csts 12392   Basecbs 12394   ↾_s cress 12395

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-ress 12402

This theorem is referenced by:  ressid  12456