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Theorem setsidvald 17236
Description: Value of the structure replacement function, deduction version.

Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (𝐸‘ndx) can be used instead of 𝑁. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 17-Oct-2024.)

Hypotheses
Ref Expression
setsidvald.e 𝐸 = Slot 𝑁
setsidvald.s (𝜑𝑆𝑉)
setsidvald.f (𝜑 → Fun 𝑆)
setsidvald.d (𝜑𝑁 ∈ dom 𝑆)
Assertion
Ref Expression
setsidvald (𝜑𝑆 = (𝑆 sSet ⟨𝑁, (𝐸𝑆)⟩))

Proof of Theorem setsidvald
StepHypRef Expression
1 setsidvald.s . . 3 (𝜑𝑆𝑉)
2 fvex 6919 . . 3 (𝐸𝑆) ∈ V
3 setsval 17204 . . 3 ((𝑆𝑉 ∧ (𝐸𝑆) ∈ V) → (𝑆 sSet ⟨𝑁, (𝐸𝑆)⟩) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸𝑆)⟩}))
41, 2, 3sylancl 586 . 2 (𝜑 → (𝑆 sSet ⟨𝑁, (𝐸𝑆)⟩) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸𝑆)⟩}))
5 setsidvald.e . . . . . 6 𝐸 = Slot 𝑁
65, 1strfvnd 17222 . . . . 5 (𝜑 → (𝐸𝑆) = (𝑆𝑁))
76opeq2d 4880 . . . 4 (𝜑 → ⟨𝑁, (𝐸𝑆)⟩ = ⟨𝑁, (𝑆𝑁)⟩)
87sneqd 4638 . . 3 (𝜑 → {⟨𝑁, (𝐸𝑆)⟩} = {⟨𝑁, (𝑆𝑁)⟩})
98uneq2d 4168 . 2 (𝜑 → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸𝑆)⟩}) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆𝑁)⟩}))
10 setsidvald.f . . 3 (𝜑 → Fun 𝑆)
11 setsidvald.d . . 3 (𝜑𝑁 ∈ dom 𝑆)
12 funresdfunsn 7209 . . 3 ((Fun 𝑆𝑁 ∈ dom 𝑆) → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆𝑁)⟩}) = 𝑆)
1310, 11, 12syl2anc 584 . 2 (𝜑 → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆𝑁)⟩}) = 𝑆)
144, 9, 133eqtrrd 2782 1 (𝜑𝑆 = (𝑆 sSet ⟨𝑁, (𝐸𝑆)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  cun 3949  {csn 4626  cop 4632  dom cdm 5685  cres 5687  Fun wfun 6555  cfv 6561  (class class class)co 7431   sSet csts 17200  Slot cslot 17218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sets 17201  df-slot 17219
This theorem is referenced by:  ressval3d  17292  opprabs  33510
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