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Theorem ressval3d 17292
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 = (𝑆s 𝐴)
ressval3d.b 𝐵 = (Base‘𝑆)
ressval3d.e 𝐸 = (Base‘ndx)
ressval3d.s (𝜑𝑆𝑉)
ressval3d.f (𝜑 → Fun 𝑆)
ressval3d.d (𝜑𝐸 ∈ dom 𝑆)
ressval3d.u (𝜑𝐴𝐵)
Assertion
Ref Expression
ressval3d (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Proof of Theorem ressval3d
StepHypRef Expression
1 ressval3d.u . 2 (𝜑𝐴𝐵)
2 sspss 4112 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
3 dfpss3 4099 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
43orbi1i 913 . . . 4 ((𝐴𝐵𝐴 = 𝐵) ↔ ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵))
52, 4bitri 275 . . 3 (𝐴𝐵 ↔ ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵))
6 simplr 769 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ¬ 𝐵𝐴)
7 ressval3d.s . . . . . . . 8 (𝜑𝑆𝑉)
87adantl 481 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑆𝑉)
9 simpl 482 . . . . . . . 8 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → 𝐴𝐵)
10 ressval3d.b . . . . . . . . . 10 𝐵 = (Base‘𝑆)
1110fvexi 6921 . . . . . . . . 9 𝐵 ∈ V
1211a1i 11 . . . . . . . 8 (𝜑𝐵 ∈ V)
13 ssexg 5329 . . . . . . . 8 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
149, 12, 13syl2an 596 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐴 ∈ V)
15 ressval3d.r . . . . . . . 8 𝑅 = (𝑆s 𝐴)
1615, 10ressval2 17279 . . . . . . 7 ((¬ 𝐵𝐴𝑆𝑉𝐴 ∈ V) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
176, 8, 14, 16syl3anc 1370 . . . . . 6 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
18 ressval3d.e . . . . . . . . . 10 𝐸 = (Base‘ndx)
1918a1i 11 . . . . . . . . 9 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐸 = (Base‘ndx))
20 dfss2 3981 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2120biimpi 216 . . . . . . . . . . . 12 (𝐴𝐵 → (𝐴𝐵) = 𝐴)
2221eqcomd 2741 . . . . . . . . . . 11 (𝐴𝐵𝐴 = (𝐴𝐵))
2322adantr 480 . . . . . . . . . 10 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → 𝐴 = (𝐴𝐵))
2423adantr 480 . . . . . . . . 9 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐴 = (𝐴𝐵))
2519, 24opeq12d 4886 . . . . . . . 8 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴𝐵)⟩)
2625eqcomd 2741 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ⟨(Base‘ndx), (𝐴𝐵)⟩ = ⟨𝐸, 𝐴⟩)
2726oveq2d 7447 . . . . . 6 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2817, 27eqtrd 2775 . . . . 5 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2928ex 412 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
3015a1i 11 . . . . . . 7 ((𝐴 = 𝐵𝜑) → 𝑅 = (𝑆s 𝐴))
31 oveq2 7439 . . . . . . . 8 (𝐴 = 𝐵 → (𝑆s 𝐴) = (𝑆s 𝐵))
3231adantr 480 . . . . . . 7 ((𝐴 = 𝐵𝜑) → (𝑆s 𝐴) = (𝑆s 𝐵))
337adantl 481 . . . . . . . 8 ((𝐴 = 𝐵𝜑) → 𝑆𝑉)
3410ressid 17290 . . . . . . . 8 (𝑆𝑉 → (𝑆s 𝐵) = 𝑆)
3533, 34syl 17 . . . . . . 7 ((𝐴 = 𝐵𝜑) → (𝑆s 𝐵) = 𝑆)
3630, 32, 353eqtrd 2779 . . . . . 6 ((𝐴 = 𝐵𝜑) → 𝑅 = 𝑆)
37 baseid 17248 . . . . . . . 8 Base = Slot (Base‘ndx)
38 ressval3d.f . . . . . . . 8 (𝜑 → Fun 𝑆)
39 ressval3d.d . . . . . . . . 9 (𝜑𝐸 ∈ dom 𝑆)
4018, 39eqeltrrid 2844 . . . . . . . 8 (𝜑 → (Base‘ndx) ∈ dom 𝑆)
4137, 7, 38, 40setsidvald 17233 . . . . . . 7 (𝜑𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
4241adantl 481 . . . . . 6 ((𝐴 = 𝐵𝜑) → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
4318a1i 11 . . . . . . . . 9 ((𝐴 = 𝐵𝜑) → 𝐸 = (Base‘ndx))
44 simpl 482 . . . . . . . . . 10 ((𝐴 = 𝐵𝜑) → 𝐴 = 𝐵)
4544, 10eqtrdi 2791 . . . . . . . . 9 ((𝐴 = 𝐵𝜑) → 𝐴 = (Base‘𝑆))
4643, 45opeq12d 4886 . . . . . . . 8 ((𝐴 = 𝐵𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (Base‘𝑆)⟩)
4746eqcomd 2741 . . . . . . 7 ((𝐴 = 𝐵𝜑) → ⟨(Base‘ndx), (Base‘𝑆)⟩ = ⟨𝐸, 𝐴⟩)
4847oveq2d 7447 . . . . . 6 ((𝐴 = 𝐵𝜑) → (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
4936, 42, 483eqtrd 2779 . . . . 5 ((𝐴 = 𝐵𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
5049ex 412 . . . 4 (𝐴 = 𝐵 → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
5129, 50jaoi 857 . . 3 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵) → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
525, 51sylbi 217 . 2 (𝐴𝐵 → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
531, 52mpcom 38 1 (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963  wpss 3964  cop 4637  dom cdm 5689  Fun wfun 6557  cfv 6563  (class class class)co 7431   sSet csts 17197  ndxcnx 17227  Basecbs 17245  s cress 17274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275
This theorem is referenced by:  estrres  18195
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