Theorem setsidvald 17138

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 6855	. . 3 ⊢ (𝐸‘𝑆) ∈ V
3		setsval 17106	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐸‘𝑆) ∈ V) → (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸‘𝑆)⟩}))
4	1, 2, 3	sylancl 587	. 2 ⊢ (𝜑 → (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝐸‘𝑆)⟩}))
5		setsidvald.e	. . . . . 6 ⊢ 𝐸 = Slot 𝑁
6	5, 1	strfvnd 17124	. . . . 5 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁))
7	6	opeq2d 4838	. . . 4 ⊢ (𝜑 → ⟨𝑁, (𝐸‘𝑆)⟩ = ⟨𝑁, (𝑆‘𝑁)⟩)
8	7	sneqd 4594	. . 3 ⊢ (𝜑 → {⟨𝑁, (𝐸‘𝑆)⟩} = {⟨𝑁, (𝑆‘𝑁)⟩})
9	8	uneq2d 4122	. 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 7145	. . 3 ⊢ ((Fun 𝑆 ∧ 𝑁 ∈ dom 𝑆) → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆‘𝑁)⟩}) = 𝑆)
13	10, 11, 12	syl2anc 585	. 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {⟨𝑁, (𝑆‘𝑁)⟩}) = 𝑆)
14	4, 9, 13	3eqtrrd 2777	1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901  {csn 4582  ⟨cop 4588  dom cdm 5632   ↾ cres 5634  Fun wfun 6494  ‘cfv 6500  (class class class)co 7368   sSet csts 17102  Slot cslot 17120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-sets 17103  df-slot 17121

This theorem is referenced by:  ressval3d  17185  opprabs  33574