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	⊢ (𝜑 → 𝑆 ∈ 𝑉)
setsidvald.f	⊢ (𝜑 → Fun 𝑆)
setsidvald.d	⊢ (𝜑 → 𝑁 ∈ dom 𝑆)

Assertion

Ref	Expression
setsidvald	⊢ (𝜑 → 𝑆 = (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩))

Proof of Theorem setsidvald

Step	Hyp	Ref	Expression
1		setsidvald.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
2		fvex 6919	. . 3 ⊢ (𝐸‘𝑆) ∈ V
3		setsval 17204	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐸‘𝑆) ∈ V) → (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸‘𝑆)⟩}))
4	1, 2, 3	sylancl 586	. 2 ⊢ (𝜑 → (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸‘𝑆)⟩}))
5		setsidvald.e	. . . . . 6 ⊢ 𝐸 = Slot 𝑁
6	5, 1	strfvnd 17222	. . . . 5 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁))
7	6	opeq2d 4880	. . . 4 ⊢ (𝜑 → ⟨𝑁, (𝐸‘𝑆)⟩ = ⟨𝑁, (𝑆‘𝑁)⟩)
8	7	sneqd 4638	. . 3 ⊢ (𝜑 → {⟨𝑁, (𝐸‘𝑆)⟩} = {⟨𝑁, (𝑆‘𝑁)⟩})
9	8	uneq2d 4168	. 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸‘𝑆)⟩}) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆‘𝑁)⟩}))
10		setsidvald.f	. . 3 ⊢ (𝜑 → Fun 𝑆)
11		setsidvald.d	. . 3 ⊢ (𝜑 → 𝑁 ∈ dom 𝑆)
12		funresdfunsn 7209	. . 3 ⊢ ((Fun 𝑆 ∧ 𝑁 ∈ dom 𝑆) → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆‘𝑁)⟩}) = 𝑆)
13	10, 11, 12	syl2anc 584	. 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆‘𝑁)⟩}) = 𝑆)
14	4, 9, 13	3eqtrrd 2782	1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩))

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