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Theorem ressid2 11607
 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 945 . . . . 5 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑊𝑋)
32elexd 2633 . . . 4 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑊 ∈ V)
4 simp3 946 . . . . 5 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝐴𝑌)
54elexd 2633 . . . 4 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝐴 ∈ V)
6 simp1 944 . . . . . 6 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝐵𝐴)
76iftrued 3404 . . . . 5 ((𝐵𝐴𝑊𝑋𝐴𝑌) → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) = 𝑊)
87, 3eqeltrd 2165 . . . 4 ((𝐵𝐴𝑊𝑋𝐴𝑌) → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) ∈ V)
9 simpl 108 . . . . . . . . 9 ((𝑤 = 𝑊𝑎 = 𝐴) → 𝑤 = 𝑊)
109fveq2d 5322 . . . . . . . 8 ((𝑤 = 𝑊𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
11 ressbas.b . . . . . . . 8 𝐵 = (Base‘𝑊)
1210, 11syl6eqr 2139 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
13 simpr 109 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → 𝑎 = 𝐴)
1412, 13sseq12d 3056 . . . . . 6 ((𝑤 = 𝑊𝑎 = 𝐴) → ((Base‘𝑤) ⊆ 𝑎𝐵𝐴))
1513, 12ineq12d 3203 . . . . . . . 8 ((𝑤 = 𝑊𝑎 = 𝐴) → (𝑎 ∩ (Base‘𝑤)) = (𝐴𝐵))
1615opeq2d 3635 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝐴𝐵)⟩)
179, 16oveq12d 5684 . . . . . 6 ((𝑤 = 𝑊𝑎 = 𝐴) → (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
1814, 9, 17ifbieq12d 3421 . . . . 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 11556 . . . . 5 s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2018, 19ovmpt2ga 5788 . . . 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 1175 . . 3 ((𝐵𝐴𝑊𝑋𝐴𝑌) → (𝑊s 𝐴) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
2221, 7eqtrd 2121 . 2 ((𝐵𝐴𝑊𝑋𝐴𝑌) → (𝑊s 𝐴) = 𝑊)
231, 22syl5eq 2133 1 ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)
 Colors of variables: wff set class Syntax hints:   → 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  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-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-setind 4366 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-br 3852  df-opab 3906  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-ress 11556 This theorem is referenced by:  ressid  11609
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