Theorem ressval2 11608

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 945	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ 𝑋)
3	2	elexd 2633	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ V)
4		simp3 946	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝐴 ∈ 𝑌)
5	4	elexd 2633	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝐴 ∈ V)
6		simp1 944	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → ¬ 𝐵 ⊆ 𝐴)
7	6	iffalsed 3407	. . . . 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 11603	. . . . . . 7 ⊢ (Base‘ndx) ∈ ℕ
9	8	a1i 9	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (Base‘ndx) ∈ ℕ)
10		inex1g 3981	. . . . . . 7 ⊢ (𝐴 ∈ 𝑌 → (𝐴 ∩ 𝐵) ∈ V)
11	4, 10	syl 14	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝐴 ∩ 𝐵) ∈ V)
12		setsex 11580	. . . . . 6 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘ndx) ∈ ℕ ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) ∈ V)
13	2, 9, 11, 12	syl3anc 1175	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) ∈ V)
14	7, 13	eqeltrd 2165	. . . 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 5322	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
17		ressbas.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊)
18	16, 17	syl6eqr 2139	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
19		simpr 109	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
20	18, 19	sseq12d 3056	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → ((Base‘𝑤) ⊆ 𝑎 ↔ 𝐵 ⊆ 𝐴))
21	19, 18	ineq12d 3203	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (𝑎 ∩ (Base‘𝑤)) = (𝐴 ∩ 𝐵))
22	21	opeq2d 3635	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)
23	15, 22	oveq12d 5684	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
24	20, 15, 23	ifbieq12d 3421	. . . . 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 11556	. . . . 5 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
26	24, 25	ovmpt2ga 5788	. . . 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 1175	. . 3 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
28	27, 7	eqtrd 2121	. 2 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
29	1, 28	syl5eq 2133	1 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∧ w3a 925   = wceq 1290   ∈ wcel 1439  Vcvv 2620   ∩ cin 2999   ⊆ wss 3000  ifcif 3397  ⟨cop 3453  ‘cfv 5028  (class class class)co 5666  ℕcn 8476  ndxcnx 11545   sSet csts 11546  Basecbs 11548   ↾_s cress 11549

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1re 7493  ax-addrcl 7496

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-iota 4993  df-fun 5030  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-inn 8477  df-ndx 11551  df-slot 11552  df-base 11554  df-sets 11555  df-ress 11556

This theorem is referenced by: (None)