Theorem setsnidel 47737

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 3466	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ V)
3		setsnidel.n	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
4	3	necomd 2988	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
5		eldifsn 4744	. . . . 5 ⊢ (𝐶 ∈ (V ∖ {𝐴}) ↔ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴))
6	2, 4, 5	sylanbrc 584	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (V ∖ {𝐴}))
7		setsnidel.s	. . . 4 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑆)
8		setsnidel.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
9		opelres 5952	. . . . 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 714	. . 3 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})))
12		elun1 4136	. . 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 17106	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
18	15, 16, 17	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
19	14, 18	eqtrid 2784	. 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
20	13, 19	eleqtrrd 2840	1 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901  {csn 4582  ⟨cop 4588   ↾ cres 5634  (class class class)co 7368   sSet csts 17102

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-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-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-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-res 5644  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-sets 17103

This theorem is referenced by: (None)