Theorem ressval3d 12904

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 12890	. . . . . . 7 |- Base Fn _V
5	2	elexd 2785	. . . . . . 7 |- ( ph -> S e. _V )
6		funfvex 5593	. . . . . . . 8 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
7	6	funfni 5376	. . . . . . 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 2292	. . . . 5 |- ( ph -> B e. _V )
10		ressval3d.u	. . . . 5 |- ( ph -> A C_ B )
11	9, 10	ssexd 4184	. . . 4 |- ( ph -> A e. _V )
12		ressvalsets 12896	. . . 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 2250	. 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 3179	. . . . . 6 |- ( A C_ B <-> ( A i^i B ) = A )
18	10, 17	sylib 122	. . . . 5 |- ( ph -> ( A i^i B ) = A )
19	3	ineq2i 3371	. . . . 5 |- ( A i^i B ) = ( A i^i ( Base ` S ) )
20	18, 19	eqtr3di 2253	. . . 4 |- ( ph -> A = ( A i^i ( Base ` S ) ) )
21	16, 20	opeq12d 3827	. . 3 |- ( ph -> <. E , A >. = <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. )
22	21	oveq2d 5960	. 2 |- ( ph -> ( S sSet <. E , A >. ) = ( S sSet <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. ) )
23	14, 22	eqtr4d 2241	1 |- ( ph -> R = ( S sSet <. E , A >. ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   _Vcvv 2772   i^i cin 3165   C_ wss 3166   <.cop 3636   dom cdm 4675   Fun wfun 5265   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   ndxcnx 12829   sSet csts 12830   Basecbs 12832   ↾_s cress 12833

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840

This theorem is referenced by: (None)