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Theorem setsidvald 17233
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 6920 . . 3 (𝐸𝑆) ∈ V
3 setsval 17201 . . 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 17219 . . . . 5 (𝜑 → (𝐸𝑆) = (𝑆𝑁))
76opeq2d 4885 . . . 4 (𝜑 → ⟨𝑁, (𝐸𝑆)⟩ = ⟨𝑁, (𝑆𝑁)⟩)
87sneqd 4643 . . 3 (𝜑 → {⟨𝑁, (𝐸𝑆)⟩} = {⟨𝑁, (𝑆𝑁)⟩})
98uneq2d 4178 . 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 2780 1 (𝜑𝑆 = (𝑆 sSet ⟨𝑁, (𝐸𝑆)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  cun 3961  {csn 4631  cop 4637  dom cdm 5689  cres 5691  Fun wfun 6557  cfv 6563  (class class class)co 7431   sSet csts 17197  Slot cslot 17215
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-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  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-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-sets 17198  df-slot 17216
This theorem is referenced by:  ressval3d  17292  opprabs  33490
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