Theorem ressval3d 15937

Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.)

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.a	|- ( ph -> A e. Y )
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.u	. 2 |- ( ph -> A C_ B )
2		sspss 3706	. . . 4 |- ( A C_ B <-> ( A C. B \/ A = B ) )
3		dfpss3 3693	. . . . 5 |- ( A C. B <-> ( A C_ B /\ -. B C_ A ) )
4	3	orbi1i 542	. . . 4 |- ( ( A C. B \/ A = B ) <-> ( ( A C_ B /\ -. B C_ A ) \/ A = B ) )
5	2, 4	bitri 264	. . 3 |- ( A C_ B <-> ( ( A C_ B /\ -. B C_ A ) \/ A = B ) )
6		simplr 792	. . . . . . 7 |- ( ( ( A C_ B /\ -. B C_ A ) /\ ph ) -> -. B C_ A )
7		ressval3d.s	. . . . . . . 8 |- ( ph -> S e. V )
8	7	adantl 482	. . . . . . 7 |- ( ( ( A C_ B /\ -. B C_ A ) /\ ph ) -> S e. V )
9		simpl 473	. . . . . . . 8 |- ( ( A C_ B /\ -. B C_ A ) -> A C_ B )
10		ressval3d.b	. . . . . . . . . 10 |- B = ( Base ` S )
11		fvex 6201	. . . . . . . . . 10 |- ( Base ` S ) e. _V
12	10, 11	eqeltri 2697	. . . . . . . . 9 |- B e. _V
13	12	a1i 11	. . . . . . . 8 |- ( ph -> B e. _V )
14		ssexg 4804	. . . . . . . 8 |- ( ( A C_ B /\ B e. _V ) -> A e. _V )
15	9, 13, 14	syl2an 494	. . . . . . 7 |- ( ( ( A C_ B /\ -. B C_ A ) /\ ph ) -> A e. _V )
16		ressval3d.r	. . . . . . . 8 |- R = ( S ↾s A )
17	16, 10	ressval2 15929	. . . . . . 7 |- ( ( -. B C_ A /\ S e. V /\ A e. _V ) -> R = ( S sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) )
18	6, 8, 15, 17	syl3anc 1326	. . . . . 6 |- ( ( ( A C_ B /\ -. B C_ A ) /\ ph ) -> R = ( S sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) )
19		ressval3d.e	. . . . . . . . . 10 |- E = ( Base ` ndx )
20	19	a1i 11	. . . . . . . . 9 |- ( ( ( A C_ B /\ -. B C_ A ) /\ ph ) -> E = ( Base ` ndx ) )
21		df-ss 3588	. . . . . . . . . . . . 13 |- ( A C_ B <-> ( A i^i B ) = A )
22	21	biimpi 206	. . . . . . . . . . . 12 |- ( A C_ B -> ( A i^i B ) = A )
23	22	eqcomd 2628	. . . . . . . . . . 11 |- ( A C_ B -> A = ( A i^i B ) )
24	23	adantr 481	. . . . . . . . . 10 |- ( ( A C_ B /\ -. B C_ A ) -> A = ( A i^i B ) )
25	24	adantr 481	. . . . . . . . 9 |- ( ( ( A C_ B /\ -. B C_ A ) /\ ph ) -> A = ( A i^i B ) )
26	20, 25	opeq12d 4410	. . . . . . . 8 |- ( ( ( A C_ B /\ -. B C_ A ) /\ ph ) -> <. E , A >. = <. ( Base ` ndx ) , ( A i^i B ) >. )
27	26	eqcomd 2628	. . . . . . 7 |- ( ( ( A C_ B /\ -. B C_ A ) /\ ph ) -> <. ( Base ` ndx ) , ( A i^i B ) >. = <. E , A >. )
28	27	oveq2d 6666	. . . . . 6 |- ( ( ( A C_ B /\ -. B C_ A ) /\ ph ) -> ( S sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) = ( S sSet <. E , A >. ) )
29	18, 28	eqtrd 2656	. . . . 5 |- ( ( ( A C_ B /\ -. B C_ A ) /\ ph ) -> R = ( S sSet <. E , A >. ) )
30	29	ex 450	. . . 4 |- ( ( A C_ B /\ -. B C_ A ) -> ( ph -> R = ( S sSet <. E , A >. ) ) )
31	16	a1i 11	. . . . . . 7 |- ( ( A = B /\ ph ) -> R = ( S ↾s A ) )
32		oveq2 6658	. . . . . . . 8 |- ( A = B -> ( S ↾s A ) = ( S ↾s B ) )
33	32	adantr 481	. . . . . . 7 |- ( ( A = B /\ ph ) -> ( S ↾s A ) = ( S ↾s B ) )
34	7	adantl 482	. . . . . . . 8 |- ( ( A = B /\ ph ) -> S e. V )
35	10	ressid 15935	. . . . . . . 8 |- ( S e. V -> ( S ↾s B ) = S )
36	34, 35	syl 17	. . . . . . 7 |- ( ( A = B /\ ph ) -> ( S ↾s B ) = S )
37	31, 33, 36	3eqtrd 2660	. . . . . 6 |- ( ( A = B /\ ph ) -> R = S )
38		df-base 15863	. . . . . . . 8 |- Base = Slot 1
39		1nn 11031	. . . . . . . 8 |- 1 e. NN
40		ressval3d.f	. . . . . . . 8 |- ( ph -> Fun S )
41		ressval3d.d	. . . . . . . . 9 |- ( ph -> E e. dom S )
42	19, 41	syl5eqelr 2706	. . . . . . . 8 |- ( ph -> ( Base ` ndx ) e. dom S )
43	38, 39, 7, 40, 42	setsidvald 15889	. . . . . . 7 |- ( ph -> S = ( S sSet <. ( Base ` ndx ) , ( Base ` S ) >. ) )
44	43	adantl 482	. . . . . 6 |- ( ( A = B /\ ph ) -> S = ( S sSet <. ( Base ` ndx ) , ( Base ` S ) >. ) )
45	19	a1i 11	. . . . . . . . 9 |- ( ( A = B /\ ph ) -> E = ( Base ` ndx ) )
46		simpl 473	. . . . . . . . . 10 |- ( ( A = B /\ ph ) -> A = B )
47	46, 10	syl6eq 2672	. . . . . . . . 9 |- ( ( A = B /\ ph ) -> A = ( Base ` S ) )
48	45, 47	opeq12d 4410	. . . . . . . 8 |- ( ( A = B /\ ph ) -> <. E , A >. = <. ( Base ` ndx ) , ( Base ` S ) >. )
49	48	eqcomd 2628	. . . . . . 7 |- ( ( A = B /\ ph ) -> <. ( Base ` ndx ) , ( Base ` S ) >. = <. E , A >. )
50	49	oveq2d 6666	. . . . . 6 |- ( ( A = B /\ ph ) -> ( S sSet <. ( Base ` ndx ) , ( Base ` S ) >. ) = ( S sSet <. E , A >. ) )
51	37, 44, 50	3eqtrd 2660	. . . . 5 |- ( ( A = B /\ ph ) -> R = ( S sSet <. E , A >. ) )
52	51	ex 450	. . . 4 |- ( A = B -> ( ph -> R = ( S sSet <. E , A >. ) ) )
53	30, 52	jaoi 394	. . 3 |- ( ( ( A C_ B /\ -. B C_ A ) \/ A = B ) -> ( ph -> R = ( S sSet <. E , A >. ) ) )
54	5, 53	sylbi 207	. 2 |- ( A C_ B -> ( ph -> R = ( S sSet <. E , A >. ) ) )
55	1, 54	mpcom 38	1 |- ( ph -> R = ( S sSet <. E , A >. ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   C. wpss 3575   <.cop 4183   dom cdm 5114   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   1c1 9937   ndxcnx 15854   sSet csts 15855   Basecbs 15857   ↾_s cress 15858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865

This theorem is referenced by:  estrres  16779