Theorem setsidel 47390

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 5439	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	snid 4638	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩}
3		elun2 4158	. . 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 17186	. . . 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 2782	. 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
11	4, 10	eleqtrrd 2837	1 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ∖ cdif 3923   ∪ cun 3924  {csn 4601  ⟨cop 4607   ↾ cres 5656  (class class class)co 7405   sSet csts 17182

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-res 5666  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-sets 17183

This theorem is referenced by: (None)