Theorem ressval2 12478

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 993	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ 𝑋)
3	2	elexd 2743	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ V)
4		simp3 994	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝐴 ∈ 𝑌)
5	4	elexd 2743	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝐴 ∈ V)
6		simp1 992	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → ¬ 𝐵 ⊆ 𝐴)
7	6	iffalsed 3536	. . . . 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 12471	. . . . . . 7 ⊢ (Base‘ndx) ∈ ℕ
9	8	a1i 9	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (Base‘ndx) ∈ ℕ)
10		inex1g 4125	. . . . . . 7 ⊢ (𝐴 ∈ 𝑌 → (𝐴 ∩ 𝐵) ∈ V)
11	4, 10	syl 14	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝐴 ∩ 𝐵) ∈ V)
12		setsex 12448	. . . . . 6 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘ndx) ∈ ℕ ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) ∈ V)
13	2, 9, 11, 12	syl3anc 1233	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) ∈ V)
14	7, 13	eqeltrd 2247	. . . 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 5500	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
17		ressbas.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊)
18	16, 17	eqtr4di 2221	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
19		simpr 109	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
20	18, 19	sseq12d 3178	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → ((Base‘𝑤) ⊆ 𝑎 ↔ 𝐵 ⊆ 𝐴))
21	19, 18	ineq12d 3329	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (𝑎 ∩ (Base‘𝑤)) = (𝐴 ∩ 𝐵))
22	21	opeq2d 3772	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)
23	15, 22	oveq12d 5871	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
24	20, 15, 23	ifbieq12d 3552	. . . . 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 12424	. . . . 5 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
26	24, 25	ovmpoga 5982	. . . 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 1233	. . 3 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
28	27, 7	eqtrd 2203	. 2 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
29	1, 28	eqtrid 2215	1 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  Vcvv 2730   ∩ cin 3120   ⊆ wss 3121  ifcif 3526  ⟨cop 3586  ‘cfv 5198  (class class class)co 5853  ℕcn 8878  ndxcnx 12413   sSet csts 12414  Basecbs 12416   ↾_s cress 12417

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-inn 8879  df-ndx 12419  df-slot 12420  df-base 12422  df-sets 12423  df-ress 12424

This theorem is referenced by: (None)