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Theorem ressval 17182
Description: Value of structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypotheses
Ref Expression
ressbas.r 𝑅 = (π‘Š β†Ύs 𝐴)
ressbas.b 𝐡 = (Baseβ€˜π‘Š)
Assertion
Ref Expression
ressval ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)))

Proof of Theorem ressval
Dummy variables 𝑀 π‘Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ressbas.r . 2 𝑅 = (π‘Š β†Ύs 𝐴)
2 elex 3487 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
3 elex 3487 . . 3 (𝐴 ∈ π‘Œ β†’ 𝐴 ∈ V)
4 simpl 482 . . . . 5 ((π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ π‘Š ∈ V)
5 ovex 7437 . . . . 5 (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩) ∈ V
6 ifcl 4568 . . . . 5 ((π‘Š ∈ V ∧ (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩) ∈ V) β†’ if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)) ∈ V)
74, 5, 6sylancl 585 . . . 4 ((π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)) ∈ V)
8 simpl 482 . . . . . . . . 9 ((𝑀 = π‘Š ∧ π‘Ž = 𝐴) β†’ 𝑀 = π‘Š)
98fveq2d 6888 . . . . . . . 8 ((𝑀 = π‘Š ∧ π‘Ž = 𝐴) β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
10 ressbas.b . . . . . . . 8 𝐡 = (Baseβ€˜π‘Š)
119, 10eqtr4di 2784 . . . . . . 7 ((𝑀 = π‘Š ∧ π‘Ž = 𝐴) β†’ (Baseβ€˜π‘€) = 𝐡)
12 simpr 484 . . . . . . 7 ((𝑀 = π‘Š ∧ π‘Ž = 𝐴) β†’ π‘Ž = 𝐴)
1311, 12sseq12d 4010 . . . . . 6 ((𝑀 = π‘Š ∧ π‘Ž = 𝐴) β†’ ((Baseβ€˜π‘€) βŠ† π‘Ž ↔ 𝐡 βŠ† 𝐴))
1412, 11ineq12d 4208 . . . . . . . 8 ((𝑀 = π‘Š ∧ π‘Ž = 𝐴) β†’ (π‘Ž ∩ (Baseβ€˜π‘€)) = (𝐴 ∩ 𝐡))
1514opeq2d 4875 . . . . . . 7 ((𝑀 = π‘Š ∧ π‘Ž = 𝐴) β†’ ⟨(Baseβ€˜ndx), (π‘Ž ∩ (Baseβ€˜π‘€))⟩ = ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)
168, 15oveq12d 7422 . . . . . 6 ((𝑀 = π‘Š ∧ π‘Ž = 𝐴) β†’ (𝑀 sSet ⟨(Baseβ€˜ndx), (π‘Ž ∩ (Baseβ€˜π‘€))⟩) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩))
1713, 8, 16ifbieq12d 4551 . . . . 5 ((𝑀 = π‘Š ∧ π‘Ž = 𝐴) β†’ if((Baseβ€˜π‘€) βŠ† π‘Ž, 𝑀, (𝑀 sSet ⟨(Baseβ€˜ndx), (π‘Ž ∩ (Baseβ€˜π‘€))⟩)) = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)))
18 df-ress 17180 . . . . 5 β†Ύs = (𝑀 ∈ V, π‘Ž ∈ V ↦ if((Baseβ€˜π‘€) βŠ† π‘Ž, 𝑀, (𝑀 sSet ⟨(Baseβ€˜ndx), (π‘Ž ∩ (Baseβ€˜π‘€))⟩)))
1917, 18ovmpoga 7557 . . . 4 ((π‘Š ∈ V ∧ 𝐴 ∈ V ∧ if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)) ∈ V) β†’ (π‘Š β†Ύs 𝐴) = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)))
207, 19mpd3an3 1458 . . 3 ((π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)))
212, 3, 20syl2an 595 . 2 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ (π‘Š β†Ύs 𝐴) = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)))
221, 21eqtrid 2778 1 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ∩ cin 3942   βŠ† wss 3943  ifcif 4523  βŸ¨cop 4629  β€˜cfv 6536  (class class class)co 7404   sSet csts 17102  ndxcnx 17132  Basecbs 17150   β†Ύs cress 17179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-ress 17180
This theorem is referenced by:  ressid2  17183  ressval2  17184  wunress  17201  wunressOLD  17202
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