Theorem ressval2 12019

Description: Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)

Hypotheses

Ref	Expression
ressbas.r	⊢ 𝑅 = (𝑊 ↾s 𝐴)
ressbas.b	⊢ 𝐵 = (Base‘𝑊)

Assertion

Ref	Expression
ressval2	⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))

Proof of Theorem ressval2

Dummy variables 𝑤 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ressbas.r	. 2 ⊢ 𝑅 = (𝑊 ↾s 𝐴)
2		simp2 982	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ 𝑋)
3	2	elexd 2699	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ V)
4		simp3 983	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝐴 ∈ 𝑌)
5	4	elexd 2699	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝐴 ∈ V)
6		simp1 981	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → ¬ 𝐵 ⊆ 𝐴)
7	6	iffalsed 3484	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
8		basendxnn 12014	. . . . . . 7 ⊢ (Base‘ndx) ∈ ℕ
9	8	a1i 9	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (Base‘ndx) ∈ ℕ)
10		inex1g 4064	. . . . . . 7 ⊢ (𝐴 ∈ 𝑌 → (𝐴 ∩ 𝐵) ∈ V)
11	4, 10	syl 14	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝐴 ∩ 𝐵) ∈ V)
12		setsex 11991	. . . . . 6 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘ndx) ∈ ℕ ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) ∈ V)
13	2, 9, 11, 12	syl3anc 1216	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) ∈ V)
14	7, 13	eqeltrd 2216	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)) ∈ V)
15		simpl 108	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
16	15	fveq2d 5425	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
17		ressbas.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊)
18	16, 17	syl6eqr 2190	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
19		simpr 109	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
20	18, 19	sseq12d 3128	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → ((Base‘𝑤) ⊆ 𝑎 ↔ 𝐵 ⊆ 𝐴))
21	19, 18	ineq12d 3278	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (𝑎 ∩ (Base‘𝑤)) = (𝐴 ∩ 𝐵))
22	21	opeq2d 3712	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)
23	15, 22	oveq12d 5792	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
24	20, 15, 23	ifbieq12d 3498	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
25		df-ress 11967	. . . . 5 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
26	24, 25	ovmpoga 5900	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V ∧ if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)) ∈ V) → (𝑊 ↾s 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
27	3, 5, 14, 26	syl3anc 1216	. . 3 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
28	27, 7	eqtrd 2172	. 2 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
29	1, 28	syl5eq 2184	1 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  Vcvv 2686   ∩ cin 3070   ⊆ wss 3071  ifcif 3474  ⟨cop 3530  ‘cfv 5123  (class class class)co 5774  ℕcn 8720  ndxcnx 11956   sSet csts 11957  Basecbs 11959   ↾_s cress 11960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1re 7714  ax-addrcl 7717

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-inn 8721  df-ndx 11962  df-slot 11963  df-base 11965  df-sets 11966  df-ress 11967

This theorem is referenced by: (None)