Theorem setsnidel 41857

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.s	. . . 4 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑆)
2		setsnidel.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
3	2	elexd 3354	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ V)
4		setsnidel.n	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
5	4	necomd 2987	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
6		eldifsn 4462	. . . . 5 ⊢ (𝐶 ∈ (V ∖ {𝐴}) ↔ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴))
7	3, 5, 6	sylanbrc 701	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (V ∖ {𝐴}))
8		setsnidel.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
9		opelresg 5557	. . . . 5 ⊢ (𝐷 ∈ 𝑌 → (⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (⟨𝐶, 𝐷⟩ ∈ 𝑆 ∧ 𝐶 ∈ (V ∖ {𝐴}))))
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (⟨𝐶, 𝐷⟩ ∈ 𝑆 ∧ 𝐶 ∈ (V ∖ {𝐴}))))
11	1, 7, 10	mpbir2and 995	. . 3 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})))
12		elun1 3923	. . 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 16090	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
18	15, 16, 17	syl2anc 696	. . 3 ⊢ (𝜑 → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
19	14, 18	syl5eq 2806	. 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
20	13, 19	eleqtrrd 2842	1 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  Vcvv 3340   ∖ cdif 3712   ∪ cun 3713  {csn 4321  ⟨cop 4327   ↾ cres 5268  (class class class)co 6813   sSet csts 16057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-res 5278  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-sets 16066

This theorem is referenced by: (None)