Theorem sprsymrelfolem2 48099

Description: Lemma 2 for sprsymrelfo 48103. (Contributed by AV, 23-Nov-2021.)

Hypothesis

Ref	Expression
sprsymrelfo.q	⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}

Assertion

Ref	Expression
sprsymrelfolem2	⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦}))

Distinct variable groups:   𝑉,𝑞   𝑄,𝑐   𝑅,𝑎,𝑏,𝑐,𝑞,𝑥,𝑦   𝑉,𝑎,𝑏,𝑐,𝑥,𝑦   𝑊,𝑎,𝑏,𝑐

Allowed substitution hints:   𝑄(𝑥,𝑦,𝑞,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑞)

Proof of Theorem sprsymrelfolem2

Step	Hyp	Ref	Expression
1		df-br 5101	. . . . . . . 8 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2		simpl 486	. . . . . . . . . 10 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) → 𝑉 ∈ 𝑊)
3		ssel 3930	. . . . . . . . . . . . 13 ⊢ (𝑅 ⊆ (𝑉 × 𝑉) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
4	3	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
5	4	imp 410	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉))
6		opelxp 5683	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
7	5, 6	sylib 220	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
8		prelspr 48092	. . . . . . . . . 10 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → {𝑥, 𝑦} ∈ (Pairs‘𝑉))
9	2, 7, 8	syl2an2r 695	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → {𝑥, 𝑦} ∈ (Pairs‘𝑉))
10	9	ex 416	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
11	1, 10	biimtrid 244	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉)) → (𝑥𝑅𝑦 → {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
12	11	3adant3 1145	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
13	12	imp 410	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} ∈ (Pairs‘𝑉))
14		vex 3458	. . . . . . . 8 ⊢ 𝑥 ∈ V
15		vex 3458	. . . . . . . 8 ⊢ 𝑦 ∈ V
16		vex 3458	. . . . . . . 8 ⊢ 𝑎 ∈ V
17		vex 3458	. . . . . . . 8 ⊢ 𝑏 ∈ V
18	14, 15, 16, 17	preq12b 4808	. . . . . . 7 ⊢ ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)))
19		breq12 5105	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥𝑅𝑦 ↔ 𝑎𝑅𝑏))
20	19	biimpd 231	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥𝑅𝑦 → 𝑎𝑅𝑏))
21	20	com12 32	. . . . . . . . . . . 12 ⊢ (𝑥𝑅𝑦 → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑎𝑅𝑏))
22	21	adantl 485	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑎𝑅𝑏))
23	22	adantr 484	. . . . . . . . . 10 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑎𝑅𝑏))
24	23	com12 32	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎𝑅𝑏))
25		rsp2 3279	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))
26	25	ancomsd 469	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))
27	26	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)) → (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))
28	27	biimpd 231	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)) → (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
29	28	ex 416	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
30	29	3ad2ant3 1148	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
31	30	com23 86	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑦𝑅𝑥)))
32	31	imp 410	. . . . . . . . . . . 12 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑦𝑅𝑥))
33	32	adantl 485	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) ∧ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑦𝑅𝑥))
34		eleq1 2850	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ 𝑉 ↔ 𝑎 ∈ 𝑉))
35		eleq1 2850	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑏 → (𝑥 ∈ 𝑉 ↔ 𝑏 ∈ 𝑉))
36	34, 35	bi2anan9r 648	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) ↔ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)))
37		breq12 5105	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑎 ∧ 𝑥 = 𝑏) → (𝑦𝑅𝑥 ↔ 𝑎𝑅𝑏))
38	37	ancoms 462	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → (𝑦𝑅𝑥 ↔ 𝑎𝑅𝑏))
39	36, 38	imbi12d 346	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → (((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑦𝑅𝑥) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎𝑅𝑏)))
40	39	adantr 484	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) ∧ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → (((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑦𝑅𝑥) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎𝑅𝑏)))
41	33, 40	mpbid 234	. . . . . . . . . 10 ⊢ (((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) ∧ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎𝑅𝑏))
42	41	expimpd 457	. . . . . . . . 9 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎𝑅𝑏))
43	24, 42	jaoi 868	. . . . . . . 8 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)) → ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎𝑅𝑏))
44	43	com12 32	. . . . . . 7 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∨ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)) → 𝑎𝑅𝑏))
45	18, 44	biimtrid 244	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏))
46	45	ralrimivva 3205	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏))
47		sprsymrelfo.q	. . . . . . 7 ⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}
48	47	eleq2i 2854	. . . . . 6 ⊢ ({𝑥, 𝑦} ∈ 𝑄 ↔ {𝑥, 𝑦} ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)})
49		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑞 = {𝑥, 𝑦} → (𝑞 = {𝑎, 𝑏} ↔ {𝑥, 𝑦} = {𝑎, 𝑏}))
50	49	imbi1d 343	. . . . . . . 8 ⊢ (𝑞 = {𝑥, 𝑦} → ((𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
51	50	2ralbidv 3226	. . . . . . 7 ⊢ (𝑞 = {𝑥, 𝑦} → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
52	51	elrab 3650	. . . . . 6 ⊢ ({𝑥, 𝑦} ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ↔ ({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
53	48, 52	bitri 277	. . . . 5 ⊢ ({𝑥, 𝑦} ∈ 𝑄 ↔ ({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
54	13, 46, 53	sylanbrc 592	. . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} ∈ 𝑄)
55		eqidd 2763	. . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} = {𝑥, 𝑦})
56		eqeq1 2766	. . . . 5 ⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 = {𝑥, 𝑦} ↔ {𝑥, 𝑦} = {𝑥, 𝑦}))
57	56	rspcev 3581	. . . 4 ⊢ (({𝑥, 𝑦} ∈ 𝑄 ∧ {𝑥, 𝑦} = {𝑥, 𝑦}) → ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦})
58	54, 55, 57	syl2anc 593	. . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦})
59	58	ex 416	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦}))
60	47	eleq2i 2854	. . . . . 6 ⊢ (𝑐 ∈ 𝑄 ↔ 𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)})
61		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑞 = 𝑐 → (𝑞 = {𝑎, 𝑏} ↔ 𝑐 = {𝑎, 𝑏}))
62	61	imbi1d 343	. . . . . . . 8 ⊢ (𝑞 = 𝑐 → ((𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
63	62	2ralbidv 3226	. . . . . . 7 ⊢ (𝑞 = 𝑐 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
64	63	elrab 3650	. . . . . 6 ⊢ (𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ↔ (𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
65	60, 64	bitri 277	. . . . 5 ⊢ (𝑐 ∈ 𝑄 ↔ (𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
66		eleq1 2850	. . . . . . . . . . 11 ⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ (Pairs‘𝑉) ↔ {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
67		prsprel 48093	. . . . . . . . . . . 12 ⊢ (({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
68	14, 15, 67	mpanr12 715	. . . . . . . . . . 11 ⊢ ({𝑥, 𝑦} ∈ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
69	66, 68	biimtrdi 255	. . . . . . . . . 10 ⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
70	69	com12 32	. . . . . . . . 9 ⊢ (𝑐 ∈ (Pairs‘𝑉) → (𝑐 = {𝑥, 𝑦} → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
71	70	adantr 484	. . . . . . . 8 ⊢ ((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) → (𝑐 = {𝑥, 𝑦} → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
72	71	imp 410	. . . . . . 7 ⊢ (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
73		preq1 4692	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → {𝑎, 𝑏} = {𝑥, 𝑏})
74	73	eqeq2d 2773	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑥, 𝑏}))
75		breq1 5103	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎𝑅𝑏 ↔ 𝑥𝑅𝑏))
76	74, 75	imbi12d 346	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ (𝑐 = {𝑥, 𝑏} → 𝑥𝑅𝑏)))
77		preq2 4693	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → {𝑥, 𝑏} = {𝑥, 𝑦})
78	77	eqeq2d 2773	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (𝑐 = {𝑥, 𝑏} ↔ 𝑐 = {𝑥, 𝑦}))
79		breq2 5104	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (𝑥𝑅𝑏 ↔ 𝑥𝑅𝑦))
80	78, 79	imbi12d 346	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → ((𝑐 = {𝑥, 𝑏} → 𝑥𝑅𝑏) ↔ (𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦)))
81	76, 80	rspc2v 3592	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏) → (𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦)))
82	81	a1d 25	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑐 ∈ (Pairs‘𝑉) → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏) → (𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦))))
83	82	imp4c 427	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → 𝑥𝑅𝑦))
84	72, 83	mpcom 38	. . . . . 6 ⊢ (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → 𝑥𝑅𝑦)
85	84	a1d 25	. . . . 5 ⊢ (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → 𝑥𝑅𝑦))
86	65, 85	sylanb 590	. . . 4 ⊢ ((𝑐 ∈ 𝑄 ∧ 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → 𝑥𝑅𝑦))
87	86	rexlimiva 3155	. . 3 ⊢ (∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦} → ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → 𝑥𝑅𝑦))
88	87	com12 32	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦))
89	59, 88	impbid 214	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  {crab 3414  Vcvv 3454   ⊆ wss 3904  {cpr 4584  ⟨cop 4588   class class class wbr 5100   × cxp 5645  ‘cfv 6521  Pairscspr 48083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6477  df-fun 6523  df-fv 6529  df-spr 48084

This theorem is referenced by:  sprsymrelfo  48103