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Theorem sprsymrelfolem2 42068
Description: Lemma 2 for sprsymrelfo 42072. (Contributed by AV, 23-Nov-2021.)
Hypothesis
Ref Expression
sprsymrelfo.q 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}
Assertion
Ref Expression
sprsymrelfolem2 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐𝑄 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
StepHypRef Expression
1 df-br 4686 . . . . . . . 8 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2 simpl 472 . . . . . . . . . 10 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) → 𝑉𝑊)
3 ssel 3630 . . . . . . . . . . . . 13 (𝑅 ⊆ (𝑉 × 𝑉) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
43adantl 481 . . . . . . . . . . . 12 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
54imp 444 . . . . . . . . . . 11 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉))
6 opelxp 5180 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉) ↔ (𝑥𝑉𝑦𝑉))
75, 6sylib 208 . . . . . . . . . 10 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → (𝑥𝑉𝑦𝑉))
8 prelspr 42061 . . . . . . . . . 10 ((𝑉𝑊 ∧ (𝑥𝑉𝑦𝑉)) → {𝑥, 𝑦} ∈ (Pairs‘𝑉))
92, 7, 8syl2an2r 893 . . . . . . . . 9 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → {𝑥, 𝑦} ∈ (Pairs‘𝑉))
109ex 449 . . . . . . . 8 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
111, 10syl5bi 232 . . . . . . 7 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) → (𝑥𝑅𝑦 → {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
12113adant3 1101 . . . . . 6 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
1312imp 444 . . . . 5 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} ∈ (Pairs‘𝑉))
14 vex 3234 . . . . . . . 8 𝑥 ∈ V
15 vex 3234 . . . . . . . 8 𝑦 ∈ V
16 vex 3234 . . . . . . . 8 𝑎 ∈ V
17 vex 3234 . . . . . . . 8 𝑏 ∈ V
1814, 15, 16, 17preq12b 4413 . . . . . . 7 ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)))
19 breq12 4690 . . . . . . . . . . . . . 14 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑅𝑦𝑎𝑅𝑏))
2019biimpd 219 . . . . . . . . . . . . 13 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑅𝑦𝑎𝑅𝑏))
2120com12 32 . . . . . . . . . . . 12 (𝑥𝑅𝑦 → ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑎𝑅𝑏))
2221adantl 481 . . . . . . . . . . 11 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑎𝑅𝑏))
2322adantr 480 . . . . . . . . . 10 ((((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎𝑉𝑏𝑉)) → ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑎𝑅𝑏))
2423com12 32 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑏) → ((((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎𝑉𝑏𝑉)) → 𝑎𝑅𝑏))
25 rsp2 2965 . . . . . . . . . . . . . . . . . . 19 (∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥) → ((𝑥𝑉𝑦𝑉) → (𝑥𝑅𝑦𝑦𝑅𝑥)))
2625ancomsd 469 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥) → ((𝑦𝑉𝑥𝑉) → (𝑥𝑅𝑦𝑦𝑅𝑥)))
2726imp 444 . . . . . . . . . . . . . . . . 17 ((∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥) ∧ (𝑦𝑉𝑥𝑉)) → (𝑥𝑅𝑦𝑦𝑅𝑥))
2827biimpd 219 . . . . . . . . . . . . . . . 16 ((∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥) ∧ (𝑦𝑉𝑥𝑉)) → (𝑥𝑅𝑦𝑦𝑅𝑥))
2928ex 449 . . . . . . . . . . . . . . 15 (∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥) → ((𝑦𝑉𝑥𝑉) → (𝑥𝑅𝑦𝑦𝑅𝑥)))
30293ad2ant3 1104 . . . . . . . . . . . . . 14 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → ((𝑦𝑉𝑥𝑉) → (𝑥𝑅𝑦𝑦𝑅𝑥)))
3130com23 86 . . . . . . . . . . . . 13 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → ((𝑦𝑉𝑥𝑉) → 𝑦𝑅𝑥)))
3231imp 444 . . . . . . . . . . . 12 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ((𝑦𝑉𝑥𝑉) → 𝑦𝑅𝑥))
3332adantl 481 . . . . . . . . . . 11 (((𝑥 = 𝑏𝑦 = 𝑎) ∧ ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → ((𝑦𝑉𝑥𝑉) → 𝑦𝑅𝑥))
34 eleq1 2718 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → (𝑦𝑉𝑎𝑉))
35 eleq1 2718 . . . . . . . . . . . . . 14 (𝑥 = 𝑏 → (𝑥𝑉𝑏𝑉))
3634, 35bi2anan9r 936 . . . . . . . . . . . . 13 ((𝑥 = 𝑏𝑦 = 𝑎) → ((𝑦𝑉𝑥𝑉) ↔ (𝑎𝑉𝑏𝑉)))
37 breq12 4690 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑥 = 𝑏) → (𝑦𝑅𝑥𝑎𝑅𝑏))
3837ancoms 468 . . . . . . . . . . . . 13 ((𝑥 = 𝑏𝑦 = 𝑎) → (𝑦𝑅𝑥𝑎𝑅𝑏))
3936, 38imbi12d 333 . . . . . . . . . . . 12 ((𝑥 = 𝑏𝑦 = 𝑎) → (((𝑦𝑉𝑥𝑉) → 𝑦𝑅𝑥) ↔ ((𝑎𝑉𝑏𝑉) → 𝑎𝑅𝑏)))
4039adantr 480 . . . . . . . . . . 11 (((𝑥 = 𝑏𝑦 = 𝑎) ∧ ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → (((𝑦𝑉𝑥𝑉) → 𝑦𝑅𝑥) ↔ ((𝑎𝑉𝑏𝑉) → 𝑎𝑅𝑏)))
4133, 40mpbid 222 . . . . . . . . . 10 (((𝑥 = 𝑏𝑦 = 𝑎) ∧ ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → ((𝑎𝑉𝑏𝑉) → 𝑎𝑅𝑏))
4241expimpd 628 . . . . . . . . 9 ((𝑥 = 𝑏𝑦 = 𝑎) → ((((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎𝑉𝑏𝑉)) → 𝑎𝑅𝑏))
4324, 42jaoi 393 . . . . . . . 8 (((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)) → ((((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎𝑉𝑏𝑉)) → 𝑎𝑅𝑏))
4443com12 32 . . . . . . 7 ((((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎𝑉𝑏𝑉)) → (((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)) → 𝑎𝑅𝑏))
4518, 44syl5bi 232 . . . . . 6 ((((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎𝑉𝑏𝑉)) → ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏))
4645ralrimivva 3000 . . . . 5 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ∀𝑎𝑉𝑏𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏))
47 sprsymrelfo.q . . . . . . 7 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}
4847eleq2i 2722 . . . . . 6 ({𝑥, 𝑦} ∈ 𝑄 ↔ {𝑥, 𝑦} ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)})
49 eqeq1 2655 . . . . . . . . 9 (𝑞 = {𝑥, 𝑦} → (𝑞 = {𝑎, 𝑏} ↔ {𝑥, 𝑦} = {𝑎, 𝑏}))
5049imbi1d 330 . . . . . . . 8 (𝑞 = {𝑥, 𝑦} → ((𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
51502ralbidv 3018 . . . . . . 7 (𝑞 = {𝑥, 𝑦} → (∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ∀𝑎𝑉𝑏𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
5251elrab 3396 . . . . . 6 ({𝑥, 𝑦} ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ↔ ({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
5348, 52bitri 264 . . . . 5 ({𝑥, 𝑦} ∈ 𝑄 ↔ ({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
5413, 46, 53sylanbrc 699 . . . 4 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} ∈ 𝑄)
55 eqidd 2652 . . . 4 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} = {𝑥, 𝑦})
56 eqeq1 2655 . . . . 5 (𝑐 = {𝑥, 𝑦} → (𝑐 = {𝑥, 𝑦} ↔ {𝑥, 𝑦} = {𝑥, 𝑦}))
5756rspcev 3340 . . . 4 (({𝑥, 𝑦} ∈ 𝑄 ∧ {𝑥, 𝑦} = {𝑥, 𝑦}) → ∃𝑐𝑄 𝑐 = {𝑥, 𝑦})
5854, 55, 57syl2anc 694 . . 3 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ∃𝑐𝑄 𝑐 = {𝑥, 𝑦})
5958ex 449 . 2 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → ∃𝑐𝑄 𝑐 = {𝑥, 𝑦}))
6047eleq2i 2722 . . . . . 6 (𝑐𝑄𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)})
61 eqeq1 2655 . . . . . . . . 9 (𝑞 = 𝑐 → (𝑞 = {𝑎, 𝑏} ↔ 𝑐 = {𝑎, 𝑏}))
6261imbi1d 330 . . . . . . . 8 (𝑞 = 𝑐 → ((𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
63622ralbidv 3018 . . . . . . 7 (𝑞 = 𝑐 → (∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
6463elrab 3396 . . . . . 6 (𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ↔ (𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
6560, 64bitri 264 . . . . 5 (𝑐𝑄 ↔ (𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
66 eleq1 2718 . . . . . . . . . . 11 (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ (Pairs‘𝑉) ↔ {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
67 prsprel 42062 . . . . . . . . . . . 12 (({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝑉𝑦𝑉))
6814, 15, 67mpanr12 721 . . . . . . . . . . 11 ({𝑥, 𝑦} ∈ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))
6966, 68syl6bi 243 . . . . . . . . . 10 (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
7069com12 32 . . . . . . . . 9 (𝑐 ∈ (Pairs‘𝑉) → (𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
7170adantr 480 . . . . . . . 8 ((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) → (𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
7271imp 444 . . . . . . 7 (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → (𝑥𝑉𝑦𝑉))
73 preq1 4300 . . . . . . . . . . . 12 (𝑎 = 𝑥 → {𝑎, 𝑏} = {𝑥, 𝑏})
7473eqeq2d 2661 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑥, 𝑏}))
75 breq1 4688 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑎𝑅𝑏𝑥𝑅𝑏))
7674, 75imbi12d 333 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ (𝑐 = {𝑥, 𝑏} → 𝑥𝑅𝑏)))
77 preq2 4301 . . . . . . . . . . . 12 (𝑏 = 𝑦 → {𝑥, 𝑏} = {𝑥, 𝑦})
7877eqeq2d 2661 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑐 = {𝑥, 𝑏} ↔ 𝑐 = {𝑥, 𝑦}))
79 breq2 4689 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑥𝑅𝑏𝑥𝑅𝑦))
8078, 79imbi12d 333 . . . . . . . . . 10 (𝑏 = 𝑦 → ((𝑐 = {𝑥, 𝑏} → 𝑥𝑅𝑏) ↔ (𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦)))
8176, 80rspc2v 3353 . . . . . . . . 9 ((𝑥𝑉𝑦𝑉) → (∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏) → (𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦)))
8281a1d 25 . . . . . . . 8 ((𝑥𝑉𝑦𝑉) → (𝑐 ∈ (Pairs‘𝑉) → (∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏) → (𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦))))
8382imp4c 616 . . . . . . 7 ((𝑥𝑉𝑦𝑉) → (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → 𝑥𝑅𝑦))
8472, 83mpcom 38 . . . . . 6 (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → 𝑥𝑅𝑦)
8584a1d 25 . . . . 5 (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → 𝑥𝑅𝑦))
8665, 85sylanb 488 . . . 4 ((𝑐𝑄𝑐 = {𝑥, 𝑦}) → ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → 𝑥𝑅𝑦))
8786rexlimiva 3057 . . 3 (∃𝑐𝑄 𝑐 = {𝑥, 𝑦} → ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → 𝑥𝑅𝑦))
8887com12 32 . 2 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (∃𝑐𝑄 𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦))
8959, 88impbid 202 1 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐𝑄 𝑐 = {𝑥, 𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  {cpr 4212  cop 4216   class class class wbr 4685   × cxp 5141  cfv 5926  Pairscspr 42052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-spr 42053
This theorem is referenced by:  sprsymrelfo  42072
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