Theorem ressval3d 12989

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	⊢ (𝜑 → 𝑆 ∈ 𝑉)
ressval3d.f	⊢ (𝜑 → Fun 𝑆)
ressval3d.d	⊢ (𝜑 → 𝐸 ∈ dom 𝑆)
ressval3d.u	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
ressval3d	⊢ (𝜑 → 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 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
3		ressval3d.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
4		basfn 12975	. . . . . . 7 ⊢ Base Fn V
5	2	elexd 2787	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V)
6		funfvex 5611	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
7	6	funfni 5390	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
8	4, 5, 7	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
9	3, 8	eqeltrid 2293	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
10		ressval3d.u	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
11	9, 10	ssexd 4195	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12		ressvalsets 12981	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝑆 ↾s 𝐴) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
13	2, 11, 12	syl2anc 411	. . 3 ⊢ (𝜑 → (𝑆 ↾s 𝐴) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
14	1, 13	eqtrid 2251	. 2 ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
15		ressval3d.e	. . . . 5 ⊢ 𝐸 = (Base‘ndx)
16	15	a1i 9	. . . 4 ⊢ (𝜑 → 𝐸 = (Base‘ndx))
17		df-ss 3183	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
18	10, 17	sylib 122	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐴)
19	3	ineq2i 3375	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ (Base‘𝑆))
20	18, 19	eqtr3di 2254	. . . 4 ⊢ (𝜑 → 𝐴 = (𝐴 ∩ (Base‘𝑆)))
21	16, 20	opeq12d 3836	. . 3 ⊢ (𝜑 → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩)
22	21	oveq2d 5978	. 2 ⊢ (𝜑 → (𝑆 sSet ⟨𝐸, 𝐴⟩) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
23	14, 22	eqtr4d 2242	1 ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2177  Vcvv 2773   ∩ cin 3169   ⊆ wss 3170  ⟨cop 3641  dom cdm 4688  Fun wfun 5279   Fn wfn 5280  ‘cfv 5285  (class class class)co 5962  ndxcnx 12914   sSet csts 12915  Basecbs 12917   ↾_s cress 12918

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1re 8049  ax-addrcl 8052

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-iota 5246  df-fun 5287  df-fn 5288  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-inn 9067  df-ndx 12920  df-slot 12921  df-base 12923  df-sets 12924  df-iress 12925

This theorem is referenced by: (None)