ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ressval2 GIF version

Theorem ressval2 12033
Description: Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypotheses
Ref Expression
ressbas.r 𝑅 = (𝑊s 𝐴)
ressbas.b 𝐵 = (Base‘𝑊)
Assertion
Ref Expression
ressval2 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))

Proof of Theorem ressval2
Dummy variables 𝑤 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ressbas.r . 2 𝑅 = (𝑊s 𝐴)
2 simp2 982 . . . . 5 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑊𝑋)
32elexd 2699 . . . 4 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑊 ∈ V)
4 simp3 983 . . . . 5 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝐴𝑌)
54elexd 2699 . . . 4 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝐴 ∈ V)
6 simp1 981 . . . . . 6 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → ¬ 𝐵𝐴)
76iffalsed 3484 . . . . 5 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
8 basendxnn 12028 . . . . . . 7 (Base‘ndx) ∈ ℕ
98a1i 9 . . . . . 6 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → (Base‘ndx) ∈ ℕ)
10 inex1g 4064 . . . . . . 7 (𝐴𝑌 → (𝐴𝐵) ∈ V)
114, 10syl 14 . . . . . 6 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → (𝐴𝐵) ∈ V)
12 setsex 12005 . . . . . 6 ((𝑊𝑋 ∧ (Base‘ndx) ∈ ℕ ∧ (𝐴𝐵) ∈ V) → (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) ∈ V)
132, 9, 11, 12syl3anc 1216 . . . . 5 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) ∈ V)
147, 13eqeltrd 2216 . . . 4 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) ∈ V)
15 simpl 108 . . . . . . . . 9 ((𝑤 = 𝑊𝑎 = 𝐴) → 𝑤 = 𝑊)
1615fveq2d 5425 . . . . . . . 8 ((𝑤 = 𝑊𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
17 ressbas.b . . . . . . . 8 𝐵 = (Base‘𝑊)
1816, 17syl6eqr 2190 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
19 simpr 109 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → 𝑎 = 𝐴)
2018, 19sseq12d 3128 . . . . . 6 ((𝑤 = 𝑊𝑎 = 𝐴) → ((Base‘𝑤) ⊆ 𝑎𝐵𝐴))
2119, 18ineq12d 3278 . . . . . . . 8 ((𝑤 = 𝑊𝑎 = 𝐴) → (𝑎 ∩ (Base‘𝑤)) = (𝐴𝐵))
2221opeq2d 3712 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝐴𝐵)⟩)
2315, 22oveq12d 5792 . . . . . 6 ((𝑤 = 𝑊𝑎 = 𝐴) → (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
2420, 15, 23ifbieq12d 3498 . . . . 5 ((𝑤 = 𝑊𝑎 = 𝐴) → if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
25 df-ress 11981 . . . . 5 s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2624, 25ovmpoga 5900 . . . 4 ((𝑊 ∈ V ∧ 𝐴 ∈ V ∧ if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) ∈ V) → (𝑊s 𝐴) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
273, 5, 14, 26syl3anc 1216 . . 3 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → (𝑊s 𝐴) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
2827, 7eqtrd 2172 . 2 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
291, 28syl5eq 2184 1 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
Colors of variables: wff set class
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  cn 8732  ndxcnx 11970   sSet csts 11971  Basecbs 11973  s cress 11974
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 7723  ax-resscn 7724  ax-1re 7726  ax-addrcl 7729
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 8733  df-ndx 11976  df-slot 11977  df-base 11979  df-sets 11980  df-ress 11981
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator