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Theorem ressval 16550
 Description: Value of 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
ressval ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))

Proof of Theorem ressval
Dummy variables 𝑤 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ressbas.r . 2 𝑅 = (𝑊s 𝐴)
2 elex 3459 . . 3 (𝑊𝑋𝑊 ∈ V)
3 elex 3459 . . 3 (𝐴𝑌𝐴 ∈ V)
4 simpl 486 . . . . 5 ((𝑊 ∈ V ∧ 𝐴 ∈ V) → 𝑊 ∈ V)
5 ovex 7173 . . . . 5 (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) ∈ V
6 ifcl 4469 . . . . 5 ((𝑊 ∈ V ∧ (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) ∈ V) → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) ∈ V)
74, 5, 6sylancl 589 . . . 4 ((𝑊 ∈ V ∧ 𝐴 ∈ V) → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) ∈ V)
8 simpl 486 . . . . . . . . 9 ((𝑤 = 𝑊𝑎 = 𝐴) → 𝑤 = 𝑊)
98fveq2d 6654 . . . . . . . 8 ((𝑤 = 𝑊𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
10 ressbas.b . . . . . . . 8 𝐵 = (Base‘𝑊)
119, 10eqtr4di 2851 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
12 simpr 488 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → 𝑎 = 𝐴)
1311, 12sseq12d 3948 . . . . . 6 ((𝑤 = 𝑊𝑎 = 𝐴) → ((Base‘𝑤) ⊆ 𝑎𝐵𝐴))
1412, 11ineq12d 4140 . . . . . . . 8 ((𝑤 = 𝑊𝑎 = 𝐴) → (𝑎 ∩ (Base‘𝑤)) = (𝐴𝐵))
1514opeq2d 4773 . . . . . . 7 ((𝑤 = 𝑊𝑎 = 𝐴) → ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝐴𝐵)⟩)
168, 15oveq12d 7158 . . . . . 6 ((𝑤 = 𝑊𝑎 = 𝐴) → (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
1713, 8, 16ifbieq12d 4452 . . . . 5 ((𝑤 = 𝑊𝑎 = 𝐴) → if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
18 df-ress 16490 . . . . 5 s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
1917, 18ovmpoga 7289 . . . 4 ((𝑊 ∈ V ∧ 𝐴 ∈ V ∧ if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) ∈ V) → (𝑊s 𝐴) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
207, 19mpd3an3 1459 . . 3 ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
212, 3, 20syl2an 598 . 2 ((𝑊𝑋𝐴𝑌) → (𝑊s 𝐴) = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
221, 21syl5eq 2845 1 ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  ifcif 4425  ⟨cop 4531  ‘cfv 6327  (class class class)co 7140  ndxcnx 16479   sSet csts 16480  Basecbs 16482   ↾s cress 16483 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6286  df-fun 6329  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-ress 16490 This theorem is referenced by:  ressid2  16551  ressval2  16552  wunress  16563
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