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Theorem ressval3d 16262
Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.)
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 3903 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
3 dfpss3 3890 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
43orbi1i 938 . . . 4 ((𝐴𝐵𝐴 = 𝐵) ↔ ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵))
52, 4bitri 267 . . 3 (𝐴𝐵 ↔ ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵))
6 simplr 786 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ¬ 𝐵𝐴)
7 ressval3d.s . . . . . . . 8 (𝜑𝑆𝑉)
87adantl 474 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑆𝑉)
9 simpl 475 . . . . . . . 8 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → 𝐴𝐵)
10 ressval3d.b . . . . . . . . . 10 𝐵 = (Base‘𝑆)
1110fvexi 6425 . . . . . . . . 9 𝐵 ∈ V
1211a1i 11 . . . . . . . 8 (𝜑𝐵 ∈ V)
13 ssexg 4999 . . . . . . . 8 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
149, 12, 13syl2an 590 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐴 ∈ V)
15 ressval3d.r . . . . . . . 8 𝑅 = (𝑆s 𝐴)
1615, 10ressval2 16254 . . . . . . 7 ((¬ 𝐵𝐴𝑆𝑉𝐴 ∈ V) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
176, 8, 14, 16syl3anc 1491 . . . . . 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 df-ss 3783 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2120biimpi 208 . . . . . . . . . . . 12 (𝐴𝐵 → (𝐴𝐵) = 𝐴)
2221eqcomd 2805 . . . . . . . . . . 11 (𝐴𝐵𝐴 = (𝐴𝐵))
2322adantr 473 . . . . . . . . . 10 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → 𝐴 = (𝐴𝐵))
2423adantr 473 . . . . . . . . 9 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐴 = (𝐴𝐵))
2519, 24opeq12d 4601 . . . . . . . 8 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴𝐵)⟩)
2625eqcomd 2805 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ⟨(Base‘ndx), (𝐴𝐵)⟩ = ⟨𝐸, 𝐴⟩)
2726oveq2d 6894 . . . . . 6 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2817, 27eqtrd 2833 . . . . 5 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2928ex 402 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
3015a1i 11 . . . . . . 7 ((𝐴 = 𝐵𝜑) → 𝑅 = (𝑆s 𝐴))
31 oveq2 6886 . . . . . . . 8 (𝐴 = 𝐵 → (𝑆s 𝐴) = (𝑆s 𝐵))
3231adantr 473 . . . . . . 7 ((𝐴 = 𝐵𝜑) → (𝑆s 𝐴) = (𝑆s 𝐵))
337adantl 474 . . . . . . . 8 ((𝐴 = 𝐵𝜑) → 𝑆𝑉)
3410ressid 16260 . . . . . . . 8 (𝑆𝑉 → (𝑆s 𝐵) = 𝑆)
3533, 34syl 17 . . . . . . 7 ((𝐴 = 𝐵𝜑) → (𝑆s 𝐵) = 𝑆)
3630, 32, 353eqtrd 2837 . . . . . 6 ((𝐴 = 𝐵𝜑) → 𝑅 = 𝑆)
37 df-base 16190 . . . . . . . 8 Base = Slot 1
38 1nn 11325 . . . . . . . 8 1 ∈ ℕ
39 ressval3d.f . . . . . . . 8 (𝜑 → Fun 𝑆)
40 ressval3d.d . . . . . . . . 9 (𝜑𝐸 ∈ dom 𝑆)
4118, 40syl5eqelr 2883 . . . . . . . 8 (𝜑 → (Base‘ndx) ∈ dom 𝑆)
4237, 38, 7, 39, 41setsidvald 16215 . . . . . . 7 (𝜑𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
4342adantl 474 . . . . . 6 ((𝐴 = 𝐵𝜑) → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
4418a1i 11 . . . . . . . . 9 ((𝐴 = 𝐵𝜑) → 𝐸 = (Base‘ndx))
45 simpl 475 . . . . . . . . . 10 ((𝐴 = 𝐵𝜑) → 𝐴 = 𝐵)
4645, 10syl6eq 2849 . . . . . . . . 9 ((𝐴 = 𝐵𝜑) → 𝐴 = (Base‘𝑆))
4744, 46opeq12d 4601 . . . . . . . 8 ((𝐴 = 𝐵𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (Base‘𝑆)⟩)
4847eqcomd 2805 . . . . . . 7 ((𝐴 = 𝐵𝜑) → ⟨(Base‘ndx), (Base‘𝑆)⟩ = ⟨𝐸, 𝐴⟩)
4948oveq2d 6894 . . . . . 6 ((𝐴 = 𝐵𝜑) → (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
5036, 43, 493eqtrd 2837 . . . . 5 ((𝐴 = 𝐵𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
5150ex 402 . . . 4 (𝐴 = 𝐵 → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
5229, 51jaoi 884 . . 3 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵) → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
535, 52sylbi 209 . 2 (𝐴𝐵 → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
541, 53mpcom 38 1 (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wo 874   = wceq 1653  wcel 2157  Vcvv 3385  cin 3768  wss 3769  wpss 3770  cop 4374  dom cdm 5312  Fun wfun 6095  cfv 6101  (class class class)co 6878  1c1 10225  ndxcnx 16181   sSet csts 16182  Basecbs 16184  s cress 16185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-1cn 10282  ax-addcl 10284
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-nn 11313  df-ndx 16187  df-slot 16188  df-base 16190  df-sets 16191  df-ress 16192
This theorem is referenced by:  estrres  17094
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