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Theorem setsidvald 17139
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 6904 . . 3 (𝐸𝑆) ∈ V
3 setsval 17107 . . 3 ((𝑆𝑉 ∧ (𝐸𝑆) ∈ V) → (𝑆 sSet ⟨𝑁, (𝐸𝑆)⟩) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸𝑆)⟩}))
41, 2, 3sylancl 585 . 2 (𝜑 → (𝑆 sSet ⟨𝑁, (𝐸𝑆)⟩) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸𝑆)⟩}))
5 setsidvald.e . . . . . 6 𝐸 = Slot 𝑁
65, 1strfvnd 17125 . . . . 5 (𝜑 → (𝐸𝑆) = (𝑆𝑁))
76opeq2d 4880 . . . 4 (𝜑 → ⟨𝑁, (𝐸𝑆)⟩ = ⟨𝑁, (𝑆𝑁)⟩)
87sneqd 4640 . . 3 (𝜑 → {⟨𝑁, (𝐸𝑆)⟩} = {⟨𝑁, (𝑆𝑁)⟩})
98uneq2d 4163 . 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 7189 . . 3 ((Fun 𝑆𝑁 ∈ dom 𝑆) → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆𝑁)⟩}) = 𝑆)
1310, 11, 12syl2anc 583 . 2 (𝜑 → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆𝑁)⟩}) = 𝑆)
144, 9, 133eqtrrd 2776 1 (𝜑𝑆 = (𝑆 sSet ⟨𝑁, (𝐸𝑆)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  Vcvv 3473  cdif 3945  cun 3946  {csn 4628  cop 4634  dom cdm 5676  cres 5678  Fun wfun 6537  cfv 6543  (class class class)co 7412   sSet csts 17103  Slot cslot 17121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-sets 17104  df-slot 17122
This theorem is referenced by:  ressval3d  17198  opprabs  33038
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