Theorem ressid2 11616

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 945	. . . . 5 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> W e. X )
3	2	elexd 2635	. . . 4 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> W e. _V )
4		simp3 946	. . . . 5 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> A e. Y )
5	4	elexd 2635	. . . 4 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> A e. _V )
6		simp1 944	. . . . . 6 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> B C_ A )
7	6	iftrued 3406	. . . . 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 2165	. . . 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 5324	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = ( Base ` W ) )
11		ressbas.b	. . . . . . . 8 |- B = ( Base ` W )
12	10, 11	syl6eqr 2139	. . . . . . 7 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = B )
13		simpr 109	. . . . . . 7 |- ( ( w = W /\ a = A ) -> a = A )
14	12, 13	sseq12d 3058	. . . . . 6 |- ( ( w = W /\ a = A ) -> ( ( Base ` w ) C_ a <-> B C_ A ) )
15	13, 12	ineq12d 3205	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( a i^i ( Base ` w ) ) = ( A i^i B ) )
16	15	opeq2d 3637	. . . . . . 7 |- ( ( w = W /\ a = A ) -> <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( A i^i B ) >. )
17	9, 16	oveq12d 5686	. . . . . 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 3423	. . . . 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 11565	. . . . 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	ovmpt2ga 5790	. . . 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 1175	. . 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 2121	. 2 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> ( W ↾s A ) = W )
23	1, 22	syl5eq 2133	1 |- ( ( B C_ A /\ W e. X /\ A e. Y ) -> R = W )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 925   = wceq 1290   e. wcel 1439   _Vcvv 2622   i^i cin 3001   C_ wss 3002   ifcif 3399   <.cop 3455   ` cfv 5030  (class class class)co 5668   ndxcnx 11554   sSet csts 11555   Basecbs 11557   ↾_s cress 11558

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-setind 4368

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2624  df-sbc 2844  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-if 3400  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-iota 4995  df-fun 5032  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-ress 11565

This theorem is referenced by:  ressid  11618