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Theorem ressval3d 17187
Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.) (Proof shortened by AV, 17-Oct-2024.)
Hypotheses
Ref Expression
ressval3d.r 𝑅 = (𝑆 β†Ύs 𝐴)
ressval3d.b 𝐡 = (Baseβ€˜π‘†)
ressval3d.e 𝐸 = (Baseβ€˜ndx)
ressval3d.s (πœ‘ β†’ 𝑆 ∈ 𝑉)
ressval3d.f (πœ‘ β†’ Fun 𝑆)
ressval3d.d (πœ‘ β†’ 𝐸 ∈ dom 𝑆)
ressval3d.u (πœ‘ β†’ 𝐴 βŠ† 𝐡)
Assertion
Ref Expression
ressval3d (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
Proof of Theorem ressval3d
StepHypRef Expression
1 ressval3d.u . 2 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
2 sspss 4098 . . . 4 (𝐴 βŠ† 𝐡 ↔ (𝐴 ⊊ 𝐡 ∨ 𝐴 = 𝐡))
3 dfpss3 4085 . . . . 5 (𝐴 ⊊ 𝐡 ↔ (𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴))
43orbi1i 912 . . . 4 ((𝐴 ⊊ 𝐡 ∨ 𝐴 = 𝐡) ↔ ((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∨ 𝐴 = 𝐡))
52, 4bitri 274 . . 3 (𝐴 βŠ† 𝐡 ↔ ((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∨ 𝐴 = 𝐡))
6 simplr 767 . . . . . . 7 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ Β¬ 𝐡 βŠ† 𝐴)
7 ressval3d.s . . . . . . . 8 (πœ‘ β†’ 𝑆 ∈ 𝑉)
87adantl 482 . . . . . . 7 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝑆 ∈ 𝑉)
9 simpl 483 . . . . . . . 8 ((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) β†’ 𝐴 βŠ† 𝐡)
10 ressval3d.b . . . . . . . . . 10 𝐡 = (Baseβ€˜π‘†)
1110fvexi 6902 . . . . . . . . 9 𝐡 ∈ V
1211a1i 11 . . . . . . . 8 (πœ‘ β†’ 𝐡 ∈ V)
13 ssexg 5322 . . . . . . . 8 ((𝐴 βŠ† 𝐡 ∧ 𝐡 ∈ V) β†’ 𝐴 ∈ V)
149, 12, 13syl2an 596 . . . . . . 7 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝐴 ∈ V)
15 ressval3d.r . . . . . . . 8 𝑅 = (𝑆 β†Ύs 𝐴)
1615, 10ressval2 17174 . . . . . . 7 ((Β¬ 𝐡 βŠ† 𝐴 ∧ 𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) β†’ 𝑅 = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩))
176, 8, 14, 16syl3anc 1371 . . . . . 6 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝑅 = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩))
18 ressval3d.e . . . . . . . . . 10 𝐸 = (Baseβ€˜ndx)
1918a1i 11 . . . . . . . . 9 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝐸 = (Baseβ€˜ndx))
20 df-ss 3964 . . . . . . . . . . . . 13 (𝐴 βŠ† 𝐡 ↔ (𝐴 ∩ 𝐡) = 𝐴)
2120biimpi 215 . . . . . . . . . . . 12 (𝐴 βŠ† 𝐡 β†’ (𝐴 ∩ 𝐡) = 𝐴)
2221eqcomd 2738 . . . . . . . . . . 11 (𝐴 βŠ† 𝐡 β†’ 𝐴 = (𝐴 ∩ 𝐡))
2322adantr 481 . . . . . . . . . 10 ((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) β†’ 𝐴 = (𝐴 ∩ 𝐡))
2423adantr 481 . . . . . . . . 9 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝐴 = (𝐴 ∩ 𝐡))
2519, 24opeq12d 4880 . . . . . . . 8 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ ⟨𝐸, 𝐴⟩ = ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)
2625eqcomd 2738 . . . . . . 7 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩ = ⟨𝐸, 𝐴⟩)
2726oveq2d 7421 . . . . . 6 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2817, 27eqtrd 2772 . . . . 5 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∧ πœ‘) β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2928ex 413 . . . 4 ((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) β†’ (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
3015a1i 11 . . . . . . 7 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝑅 = (𝑆 β†Ύs 𝐴))
31 oveq2 7413 . . . . . . . 8 (𝐴 = 𝐡 β†’ (𝑆 β†Ύs 𝐴) = (𝑆 β†Ύs 𝐡))
3231adantr 481 . . . . . . 7 ((𝐴 = 𝐡 ∧ πœ‘) β†’ (𝑆 β†Ύs 𝐴) = (𝑆 β†Ύs 𝐡))
337adantl 482 . . . . . . . 8 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝑆 ∈ 𝑉)
3410ressid 17185 . . . . . . . 8 (𝑆 ∈ 𝑉 β†’ (𝑆 β†Ύs 𝐡) = 𝑆)
3533, 34syl 17 . . . . . . 7 ((𝐴 = 𝐡 ∧ πœ‘) β†’ (𝑆 β†Ύs 𝐡) = 𝑆)
3630, 32, 353eqtrd 2776 . . . . . 6 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝑅 = 𝑆)
37 baseid 17143 . . . . . . . 8 Base = Slot (Baseβ€˜ndx)
38 ressval3d.f . . . . . . . 8 (πœ‘ β†’ Fun 𝑆)
39 ressval3d.d . . . . . . . . 9 (πœ‘ β†’ 𝐸 ∈ dom 𝑆)
4018, 39eqeltrrid 2838 . . . . . . . 8 (πœ‘ β†’ (Baseβ€˜ndx) ∈ dom 𝑆)
4137, 7, 38, 40setsidvald 17128 . . . . . . 7 (πœ‘ β†’ 𝑆 = (𝑆 sSet ⟨(Baseβ€˜ndx), (Baseβ€˜π‘†)⟩))
4241adantl 482 . . . . . 6 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝑆 = (𝑆 sSet ⟨(Baseβ€˜ndx), (Baseβ€˜π‘†)⟩))
4318a1i 11 . . . . . . . . 9 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝐸 = (Baseβ€˜ndx))
44 simpl 483 . . . . . . . . . 10 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝐴 = 𝐡)
4544, 10eqtrdi 2788 . . . . . . . . 9 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝐴 = (Baseβ€˜π‘†))
4643, 45opeq12d 4880 . . . . . . . 8 ((𝐴 = 𝐡 ∧ πœ‘) β†’ ⟨𝐸, 𝐴⟩ = ⟨(Baseβ€˜ndx), (Baseβ€˜π‘†)⟩)
4746eqcomd 2738 . . . . . . 7 ((𝐴 = 𝐡 ∧ πœ‘) β†’ ⟨(Baseβ€˜ndx), (Baseβ€˜π‘†)⟩ = ⟨𝐸, 𝐴⟩)
4847oveq2d 7421 . . . . . 6 ((𝐴 = 𝐡 ∧ πœ‘) β†’ (𝑆 sSet ⟨(Baseβ€˜ndx), (Baseβ€˜π‘†)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
4936, 42, 483eqtrd 2776 . . . . 5 ((𝐴 = 𝐡 ∧ πœ‘) β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
5049ex 413 . . . 4 (𝐴 = 𝐡 β†’ (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
5129, 50jaoi 855 . . 3 (((𝐴 βŠ† 𝐡 ∧ Β¬ 𝐡 βŠ† 𝐴) ∨ 𝐴 = 𝐡) β†’ (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
525, 51sylbi 216 . 2 (𝐴 βŠ† 𝐡 β†’ (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
531, 52mpcom 38 1 (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3946   βŠ† wss 3947   ⊊ wpss 3948  βŸ¨cop 4633  dom cdm 5675  Fun wfun 6534  β€˜cfv 6540  (class class class)co 7405   sSet csts 17092  ndxcnx 17122  Basecbs 17140   β†Ύs cress 17169
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-nn 12209  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170
This theorem is referenced by:  estrres  18087
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