Theorem setsidvald 17260

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 6896	. . 3 ⊢ (𝐸‘𝑆) ∈ V
3		setsval 17228	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐸‘𝑆) ∈ V) → (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸‘𝑆)⟩}))
4	1, 2, 3	sylancl 597	. 2 ⊢ (𝜑 → (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸‘𝑆)⟩}))
5		setsidvald.e	. . . . . 6 ⊢ 𝐸 = Slot 𝑁
6	5, 1	strfvnd 17246	. . . . 5 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁))
7	6	opeq2d 4846	. . . 4 ⊢ (𝜑 → ⟨𝑁, (𝐸‘𝑆)⟩ = ⟨𝑁, (𝑆‘𝑁)⟩)
8	7	sneqd 4602	. . 3 ⊢ (𝜑 → {⟨𝑁, (𝐸‘𝑆)⟩} = {⟨𝑁, (𝑆‘𝑁)⟩})
9	8	uneq2d 4123	. 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 7189	. . 3 ⊢ ((Fun 𝑆 ∧ 𝑁 ∈ dom 𝑆) → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆‘𝑁)⟩}) = 𝑆)
13	10, 11, 12	syl2anc 595	. 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆‘𝑁)⟩}) = 𝑆)
14	4, 9, 13	3eqtrrd 2803	1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3903   ∪ cun 3904  {csn 4590  ⟨cop 4596  dom cdm 5663   ↾ cres 5665  Fun wfun 6532  ‘cfv 6538  (class class class)co 7412   sSet csts 17224  Slot cslot 17242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-sets 17225  df-slot 17243

This theorem is referenced by:  ressval3d  17307  opprabs  33745