Theorem ressval3d 13154

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 13140	. . . . . . 7 |- Base Fn _V
5	2	elexd 2816	. . . . . . 7 |- ( ph -> S e. _V )
6		funfvex 5656	. . . . . . . 8 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
7	6	funfni 5432	. . . . . . 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 2318	. . . . 5 |- ( ph -> B e. _V )
10		ressval3d.u	. . . . 5 |- ( ph -> A C_ B )
11	9, 10	ssexd 4229	. . . 4 |- ( ph -> A e. _V )
12		ressvalsets 13146	. . . 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 2276	. 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 3213	. . . . . 6 |- ( A C_ B <-> ( A i^i B ) = A )
18	10, 17	sylib 122	. . . . 5 |- ( ph -> ( A i^i B ) = A )
19	3	ineq2i 3405	. . . . 5 |- ( A i^i B ) = ( A i^i ( Base ` S ) )
20	18, 19	eqtr3di 2279	. . . 4 |- ( ph -> A = ( A i^i ( Base ` S ) ) )
21	16, 20	opeq12d 3870	. . 3 |- ( ph -> <. E , A >. = <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. )
22	21	oveq2d 6033	. 2 |- ( ph -> ( S sSet <. E , A >. ) = ( S sSet <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. ) )
23	14, 22	eqtr4d 2267	1 |- ( ph -> R = ( S sSet <. E , A >. ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   i^i cin 3199   C_ wss 3200   <.cop 3672   dom cdm 4725   Fun wfun 5320   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   ndxcnx 13078   sSet csts 13079   Basecbs 13081   ↾_s cress 13082

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089

This theorem is referenced by: (None)