ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ressid2 GIF version

Theorem ressid2 12477
Description: General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 26-Jan-2023.)
Hypotheses
Ref Expression
ressbas.r 𝑅 = (𝑊s 𝐴)
ressbas.b 𝐵 = (Base‘𝑊)
Assertion
Ref Expression
ressid2 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)

Proof of Theorem ressid2
Dummy variables 𝑤 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ressbas.r . 2 𝑅 = (𝑊s 𝐴)
2 simp2 993 . . . . 5 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑊𝑋)
32elexd 2743 . . . 4 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑊 ∈ V)
4 simp3 994 . . . . 5 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝐴𝑌)
54elexd 2743 . . . 4 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝐴 ∈ V)
6 simp1 992 . . . . . 6 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝐵𝐴)
76iftrued 3533 . . . . 5 ((𝐵𝐴𝑊𝑋𝐴𝑌) → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) = 𝑊)
87, 3eqeltrd 2247 . . . 4 ((𝐵𝐴𝑊𝑋𝐴𝑌) → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) ∈ V)
9 simpl 108 . . . . . . . . 9 ((𝑤 = 𝑊𝑎 = 𝐴) → 𝑤 = 𝑊)
109fveq2d 5500 . . . . . . . 8 ((𝑤 = 𝑊𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
11 ressbas.b . . . . . . . 8 𝐵 = (Base‘𝑊)
1210, 11eqtr4di 2221 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
13 simpr 109 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → 𝑎 = 𝐴)
1412, 13sseq12d 3178 . . . . . 6 ((𝑤 = 𝑊𝑎 = 𝐴) → ((Base‘𝑤) ⊆ 𝑎𝐵𝐴))
1513, 12ineq12d 3329 . . . . . . . 8 ((𝑤 = 𝑊𝑎 = 𝐴) → (𝑎 ∩ (Base‘𝑤)) = (𝐴𝐵))
1615opeq2d 3772 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝐴𝐵)⟩)
179, 16oveq12d 5871 . . . . . 6 ((𝑤 = 𝑊𝑎 = 𝐴) → (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
1814, 9, 17ifbieq12d 3552 . . . . 5 ((𝑤 = 𝑊𝑎 = 𝐴) → if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
19 df-ress 12424 . . . . 5 s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2018, 19ovmpoga 5982 . . . 4 ((𝑊 ∈ V ∧ 𝐴 ∈ V ∧ if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) ∈ V) → (𝑊s 𝐴) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
213, 5, 8, 20syl3anc 1233 . . 3 ((𝐵𝐴𝑊𝑋𝐴𝑌) → (𝑊s 𝐴) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
2221, 7eqtrd 2203 . 2 ((𝐵𝐴𝑊𝑋𝐴𝑌) → (𝑊s 𝐴) = 𝑊)
231, 22eqtrid 2215 1 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)
Colors of variables: wff set class
Syntax hints:  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  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-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521
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-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-ress 12424
This theorem is referenced by:  ressid  12479
  Copyright terms: Public domain W3C validator