Theorem setsnidel 41347

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 3214	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ V)
4		setsnidel.n	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
5	4	necomd 2849	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
6		eldifsn 4317	. . . . 5 ⊢ (𝐶 ∈ (V ∖ {𝐴}) ↔ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴))
7	3, 5, 6	sylanbrc 698	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (V ∖ {𝐴}))
8		setsnidel.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
9		opelresg 5404	. . . . 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 957	. . 3 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})))
12		elun1 3780	. . 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 15888	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
18	15, 16, 17	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
19	14, 18	syl5eq 2668	. 2 ⊢ (𝜑 → 𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
20	13, 19	eleqtrrd 2704	1 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572  {csn 4177  ⟨cop 4183   ↾ cres 5116  (class class class)co 6650   sSet csts 15855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sets 15864

This theorem is referenced by: (None)