Theorem setsidel 44797

Description: The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.)

Hypotheses

Ref	Expression
setsidel.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
setsidel.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
setsidel.r	⊢ 𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩)

Assertion

Ref	Expression
setsidel	⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)

Proof of Theorem setsidel

Step	Hyp	Ref	Expression
1		opex 5383	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	snid 4603	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩}
3		elun2 4116	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝐴, 𝐵⟩ ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
4	2, 3	mp1i 13	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
5		setsidel.r	. . 3 ⊢ 𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩)
6		setsidel.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
7		setsidel.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
8		setsval 16866	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
9	6, 7, 8	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
10	5, 9	eqtrid 2792	. 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
11	4, 10	eleqtrrd 2844	1 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2110  Vcvv 3431   ∖ cdif 3889   ∪ cun 3890  {csn 4567  ⟨cop 4573   ↾ cres 5592  (class class class)co 7271   sSet csts 16862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-res 5602  df-iota 6390  df-fun 6434  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-sets 16863

This theorem is referenced by: (None)