Theorem ressval3d 12534

Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)

Hypotheses

Ref	Expression
ressval3d.r	|- R = ( S ↾s A )
ressval3d.b	|- B = ( Base ` S )
ressval3d.e	|- E = ( Base ` ndx )
ressval3d.s	|- ( ph -> S e. V )
ressval3d.f	|- ( ph -> Fun S )
ressval3d.d	|- ( ph -> E e. dom S )
ressval3d.u	|- ( ph -> A C_ B )

Assertion

Ref	Expression
ressval3d	|- ( ph -> R = ( S sSet <. E , A >. ) )

Proof of Theorem ressval3d

Step	Hyp	Ref	Expression
1		ressval3d.r	. . 3 |- R = ( S ↾s A )
2		ressval3d.s	. . . 4 |- ( ph -> S e. V )
3		ressval3d.b	. . . . . 6 |- B = ( Base ` S )
4		basfn 12523	. . . . . . 7 |- Base Fn _V
5	2	elexd 2752	. . . . . . 7 |- ( ph -> S e. _V )
6		funfvex 5534	. . . . . . . 8 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
7	6	funfni 5318	. . . . . . 7 |- ( ( Base Fn _V /\ S e. _V ) -> ( Base ` S ) e. _V )
8	4, 5, 7	sylancr 414	. . . . . 6 |- ( ph -> ( Base ` S ) e. _V )
9	3, 8	eqeltrid 2264	. . . . 5 |- ( ph -> B e. _V )
10		ressval3d.u	. . . . 5 |- ( ph -> A C_ B )
11	9, 10	ssexd 4145	. . . 4 |- ( ph -> A e. _V )
12		ressvalsets 12527	. . . 4 |- ( ( S e. V /\ A e. _V ) -> ( S ↾s A ) = ( S sSet <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. ) )
13	2, 11, 12	syl2anc 411	. . 3 |- ( ph -> ( S ↾s A ) = ( S sSet <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. ) )
14	1, 13	eqtrid 2222	. 2 |- ( ph -> R = ( S sSet <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. ) )
15		ressval3d.e	. . . . 5 |- E = ( Base ` ndx )
16	15	a1i 9	. . . 4 |- ( ph -> E = ( Base ` ndx ) )
17		df-ss 3144	. . . . . 6 |- ( A C_ B <-> ( A i^i B ) = A )
18	10, 17	sylib 122	. . . . 5 |- ( ph -> ( A i^i B ) = A )
19	3	ineq2i 3335	. . . . 5 |- ( A i^i B ) = ( A i^i ( Base ` S ) )
20	18, 19	eqtr3di 2225	. . . 4 |- ( ph -> A = ( A i^i ( Base ` S ) ) )
21	16, 20	opeq12d 3788	. . 3 |- ( ph -> <. E , A >. = <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. )
22	21	oveq2d 5894	. 2 |- ( ph -> ( S sSet <. E , A >. ) = ( S sSet <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. ) )
23	14, 22	eqtr4d 2213	1 |- ( ph -> R = ( S sSet <. E , A >. ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2739   i^i cin 3130   C_ wss 3131   <.cop 3597   dom cdm 4628   Fun wfun 5212   Fn wfn 5213   ` cfv 5218  (class class class)co 5878   ndxcnx 12462   sSet csts 12463   Basecbs 12465   ↾_s cress 12466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-inn 8923  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-iress 12473

This theorem is referenced by: (None)