Theorem ressval3d 13403

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 13389	. . . . . . 7 |- Base Fn _V
5	2	elexd 2835	. . . . . . 7 |- ( ph -> S e. _V )
6		funfvex 5707	. . . . . . . 8 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
7	6	funfni 5478	. . . . . . 7 |- ( ( Base Fn _V /\ S e. _V ) -> ( Base ` S ) e. _V )
8	4, 5, 7	sylancr 418	. . . . . 6 |- ( ph -> ( Base ` S ) e. _V )
9	3, 8	eqeltrid 2325	. . . . 5 |- ( ph -> B e. _V )
10		ressval3d.u	. . . . 5 |- ( ph -> A C_ B )
11	9, 10	ssexd 4268	. . . 4 |- ( ph -> A e. _V )
12		ressvalsets 13395	. . . 4 |- ( ( S e. V /\ A e. _V ) -> ( S ↾s A ) = ( S sSet <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. ) )
13	2, 11, 12	syl2anc 415	. . 3 |- ( ph -> ( S ↾s A ) = ( S sSet <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. ) )
14	1, 13	eqtrid 2283	. 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 3233	. . . . . 6 |- ( A C_ B <-> ( A i^i B ) = A )
18	10, 17	sylib 122	. . . . 5 |- ( ph -> ( A i^i B ) = A )
19	3	ineq2i 3429	. . . . 5 |- ( A i^i B ) = ( A i^i ( Base ` S ) )
20	18, 19	eqtr3di 2286	. . . 4 |- ( ph -> A = ( A i^i ( Base ` S ) ) )
21	16, 20	opeq12d 3907	. . 3 |- ( ph -> <. E , A >. = <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. )
22	21	oveq2d 6091	. 2 |- ( ph -> ( S sSet <. E , A >. ) = ( S sSet <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. ) )
23	14, 22	eqtr4d 2274	1 |- ( ph -> R = ( S sSet <. E , A >. ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   C_ wss 3220   <.cop 3708   dom cdm 4769   Fun wfun 5366   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   ndxcnx 13327   sSet csts 13328   Basecbs 13330   ↾_s cress 13331

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338

This theorem is referenced by: (None)