Theorem setsnidel 47364

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⟩)
setsnidel.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)
setsnidel.d	⊢ (𝜑 → 𝐷 ∈ 𝑌)
setsnidel.s	⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑆)
setsnidel.n	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Assertion

Ref	Expression
setsnidel	⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)

Proof of Theorem setsnidel

Step	Hyp	Ref	Expression
1		setsnidel.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
2	1	elexd 3504	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ V)
3		setsnidel.n	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
4	3	necomd 2996	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
5		eldifsn 4786	. . . . 5 ⊢ (𝐶 ∈ (V ∖ {𝐴}) ↔ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴))
6	2, 4, 5	sylanbrc 583	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (V ∖ {𝐴}))
7		setsnidel.s	. . . 4 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑆)
8		setsnidel.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
9		opelres 6003	. . . . 5 ⊢ (𝐷 ∈ 𝑌 → (⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (𝐶 ∈ (V ∖ {𝐴}) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑆)))
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (𝐶 ∈ (V ∖ {𝐴}) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑆)))
11	6, 7, 10	mpbir2and 713	. . 3 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})))
12		elun1 4182	. . 3 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})) → ⟨𝐶, 𝐷⟩ ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
13	11, 12	syl 17	. 2 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
14		setsidel.r	. . 3 ⊢ 𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩)
15		setsidel.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
16		setsidel.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
17		setsval 17204	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
18	15, 16, 17	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
19	14, 18	eqtrid 2789	. 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
20	13, 19	eleqtrrd 2844	1 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949  {csn 4626  ⟨cop 4632   ↾ cres 5687  (class class class)co 7431   sSet csts 17200

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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sets 17201

This theorem is referenced by: (None)