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Theorem ressval3d 16378
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 3992 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
3 dfpss3 3979 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
43orbi1i 906 . . . 4 ((𝐴𝐵𝐴 = 𝐵) ↔ ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵))
52, 4bitri 276 . . 3 (𝐴𝐵 ↔ ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵))
6 simplr 765 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ¬ 𝐵𝐴)
7 ressval3d.s . . . . . . . 8 (𝜑𝑆𝑉)
87adantl 482 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑆𝑉)
9 simpl 483 . . . . . . . 8 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → 𝐴𝐵)
10 ressval3d.b . . . . . . . . . 10 𝐵 = (Base‘𝑆)
1110fvexi 6544 . . . . . . . . 9 𝐵 ∈ V
1211a1i 11 . . . . . . . 8 (𝜑𝐵 ∈ V)
13 ssexg 5111 . . . . . . . 8 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
149, 12, 13syl2an 595 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐴 ∈ V)
15 ressval3d.r . . . . . . . 8 𝑅 = (𝑆s 𝐴)
1615, 10ressval2 16370 . . . . . . 7 ((¬ 𝐵𝐴𝑆𝑉𝐴 ∈ V) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
176, 8, 14, 16syl3anc 1362 . . . . . 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 3869 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2120biimpi 217 . . . . . . . . . . . 12 (𝐴𝐵 → (𝐴𝐵) = 𝐴)
2221eqcomd 2799 . . . . . . . . . . 11 (𝐴𝐵𝐴 = (𝐴𝐵))
2322adantr 481 . . . . . . . . . 10 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → 𝐴 = (𝐴𝐵))
2423adantr 481 . . . . . . . . 9 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐴 = (𝐴𝐵))
2519, 24opeq12d 4712 . . . . . . . 8 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴𝐵)⟩)
2625eqcomd 2799 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ⟨(Base‘ndx), (𝐴𝐵)⟩ = ⟨𝐸, 𝐴⟩)
2726oveq2d 7023 . . . . . 6 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2817, 27eqtrd 2829 . . . . 5 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2928ex 413 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
3015a1i 11 . . . . . . 7 ((𝐴 = 𝐵𝜑) → 𝑅 = (𝑆s 𝐴))
31 oveq2 7015 . . . . . . . 8 (𝐴 = 𝐵 → (𝑆s 𝐴) = (𝑆s 𝐵))
3231adantr 481 . . . . . . 7 ((𝐴 = 𝐵𝜑) → (𝑆s 𝐴) = (𝑆s 𝐵))
337adantl 482 . . . . . . . 8 ((𝐴 = 𝐵𝜑) → 𝑆𝑉)
3410ressid 16376 . . . . . . . 8 (𝑆𝑉 → (𝑆s 𝐵) = 𝑆)
3533, 34syl 17 . . . . . . 7 ((𝐴 = 𝐵𝜑) → (𝑆s 𝐵) = 𝑆)
3630, 32, 353eqtrd 2833 . . . . . 6 ((𝐴 = 𝐵𝜑) → 𝑅 = 𝑆)
37 df-base 16306 . . . . . . . 8 Base = Slot 1
38 1nn 11486 . . . . . . . 8 1 ∈ ℕ
39 ressval3d.f . . . . . . . 8 (𝜑 → Fun 𝑆)
40 ressval3d.d . . . . . . . . 9 (𝜑𝐸 ∈ dom 𝑆)
4118, 40syl5eqelr 2886 . . . . . . . 8 (𝜑 → (Base‘ndx) ∈ dom 𝑆)
4237, 38, 7, 39, 41setsidvald 16331 . . . . . . 7 (𝜑𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
4342adantl 482 . . . . . 6 ((𝐴 = 𝐵𝜑) → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
4418a1i 11 . . . . . . . . 9 ((𝐴 = 𝐵𝜑) → 𝐸 = (Base‘ndx))
45 simpl 483 . . . . . . . . . 10 ((𝐴 = 𝐵𝜑) → 𝐴 = 𝐵)
4645, 10syl6eq 2845 . . . . . . . . 9 ((𝐴 = 𝐵𝜑) → 𝐴 = (Base‘𝑆))
4744, 46opeq12d 4712 . . . . . . . 8 ((𝐴 = 𝐵𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (Base‘𝑆)⟩)
4847eqcomd 2799 . . . . . . 7 ((𝐴 = 𝐵𝜑) → ⟨(Base‘ndx), (Base‘𝑆)⟩ = ⟨𝐸, 𝐴⟩)
4948oveq2d 7023 . . . . . 6 ((𝐴 = 𝐵𝜑) → (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
5036, 43, 493eqtrd 2833 . . . . 5 ((𝐴 = 𝐵𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
5150ex 413 . . . 4 (𝐴 = 𝐵 → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
5229, 51jaoi 852 . . 3 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵) → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
535, 52sylbi 218 . 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 396  wo 842   = wceq 1520  wcel 2079  Vcvv 3432  cin 3853  wss 3854  wpss 3855  cop 4472  dom cdm 5435  Fun wfun 6211  cfv 6217  (class class class)co 7007  1c1 10373  ndxcnx 16297   sSet csts 16298  Basecbs 16300  s cress 16301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-1cn 10430  ax-addcl 10432
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-nn 11476  df-ndx 16303  df-slot 16304  df-base 16306  df-sets 16307  df-ress 16308
This theorem is referenced by:  estrres  17206
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