Theorem setsidel 47836

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 5416	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2	1	snid 4606	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩}
3		elun2 4123	. . 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 17137	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
9	6, 7, 8	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
10	5, 9	eqtrid 2783	. 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
11	4, 10	eleqtrrd 2839	1 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∖ cdif 3886   ∪ cun 3887  {csn 4567  ⟨cop 4573   ↾ cres 5633  (class class class)co 7367   sSet csts 17133

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 2708  ax-sep 5231  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-sets 17134

This theorem is referenced by: (None)