Theorem ressval3d 13285

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 13271	. . . . . . 7 |- Base Fn _V
5	2	elexd 2827	. . . . . . 7 |- ( ph -> S e. _V )
6		funfvex 5687	. . . . . . . 8 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
7	6	funfni 5458	. . . . . . 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 2319	. . . . 5 |- ( ph -> B e. _V )
10		ressval3d.u	. . . . 5 |- ( ph -> A C_ B )
11	9, 10	ssexd 4250	. . . 4 |- ( ph -> A e. _V )
12		ressvalsets 13277	. . . 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 2277	. 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 3224	. . . . . 6 |- ( A C_ B <-> ( A i^i B ) = A )
18	10, 17	sylib 122	. . . . 5 |- ( ph -> ( A i^i B ) = A )
19	3	ineq2i 3419	. . . . 5 |- ( A i^i B ) = ( A i^i ( Base ` S ) )
20	18, 19	eqtr3di 2280	. . . 4 |- ( ph -> A = ( A i^i ( Base ` S ) ) )
21	16, 20	opeq12d 3891	. . 3 |- ( ph -> <. E , A >. = <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. )
22	21	oveq2d 6066	. 2 |- ( ph -> ( S sSet <. E , A >. ) = ( S sSet <. ( Base ` ndx ) , ( A i^i ( Base ` S ) ) >. ) )
23	14, 22	eqtr4d 2268	1 |- ( ph -> R = ( S sSet <. E , A >. ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2813   i^i cin 3210   C_ wss 3211   <.cop 3692   dom cdm 4749   Fun wfun 5346   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   ndxcnx 13209   sSet csts 13210   Basecbs 13212   ↾_s cress 13213

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220

This theorem is referenced by: (None)