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Theorem sprsymrelfolem2 43801
Description: Lemma 2 for sprsymrelfo 43805. (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 5043 . . . . . . . 8 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2 simpl 485 . . . . . . . . . 10 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) → 𝑉𝑊)
3 ssel 3940 . . . . . . . . . . . . 13 (𝑅 ⊆ (𝑉 × 𝑉) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
43adantl 484 . . . . . . . . . . . 12 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
54imp 409 . . . . . . . . . . 11 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉))
6 opelxp 5567 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉) ↔ (𝑥𝑉𝑦𝑉))
75, 6sylib 220 . . . . . . . . . 10 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → (𝑥𝑉𝑦𝑉))
8 prelspr 43794 . . . . . . . . . 10 ((𝑉𝑊 ∧ (𝑥𝑉𝑦𝑉)) → {𝑥, 𝑦} ∈ (Pairs‘𝑉))
92, 7, 8syl2an2r 683 . . . . . . . . 9 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → {𝑥, 𝑦} ∈ (Pairs‘𝑉))
109ex 415 . . . . . . . 8 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
111, 10syl5bi 244 . . . . . . 7 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉)) → (𝑥𝑅𝑦 → {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
12113adant3 1128 . . . . . 6 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
1312imp 409 . . . . 5 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} ∈ (Pairs‘𝑉))
14 vex 3476 . . . . . . . 8 𝑥 ∈ V
15 vex 3476 . . . . . . . 8 𝑦 ∈ V
16 vex 3476 . . . . . . . 8 𝑎 ∈ V
17 vex 3476 . . . . . . . 8 𝑏 ∈ V
1814, 15, 16, 17preq12b 4757 . . . . . . 7 ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)))
19 breq12 5047 . . . . . . . . . . . . . 14 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑅𝑦𝑎𝑅𝑏))
2019biimpd 231 . . . . . . . . . . . . 13 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑅𝑦𝑎𝑅𝑏))
2120com12 32 . . . . . . . . . . . 12 (𝑥𝑅𝑦 → ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑎𝑅𝑏))
2221adantl 484 . . . . . . . . . . 11 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑎𝑅𝑏))
2322adantr 483 . . . . . . . . . 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 3200 . . . . . . . . . . . . . . . . . . 19 (∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥) → ((𝑥𝑉𝑦𝑉) → (𝑥𝑅𝑦𝑦𝑅𝑥)))
2625ancomsd 468 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥) → ((𝑦𝑉𝑥𝑉) → (𝑥𝑅𝑦𝑦𝑅𝑥)))
2726imp 409 . . . . . . . . . . . . . . . . 17 ((∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥) ∧ (𝑦𝑉𝑥𝑉)) → (𝑥𝑅𝑦𝑦𝑅𝑥))
2827biimpd 231 . . . . . . . . . . . . . . . 16 ((∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥) ∧ (𝑦𝑉𝑥𝑉)) → (𝑥𝑅𝑦𝑦𝑅𝑥))
2928ex 415 . . . . . . . . . . . . . . 15 (∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥) → ((𝑦𝑉𝑥𝑉) → (𝑥𝑅𝑦𝑦𝑅𝑥)))
30293ad2ant3 1131 . . . . . . . . . . . . . 14 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → ((𝑦𝑉𝑥𝑉) → (𝑥𝑅𝑦𝑦𝑅𝑥)))
3130com23 86 . . . . . . . . . . . . 13 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → ((𝑦𝑉𝑥𝑉) → 𝑦𝑅𝑥)))
3231imp 409 . . . . . . . . . . . 12 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ((𝑦𝑉𝑥𝑉) → 𝑦𝑅𝑥))
3332adantl 484 . . . . . . . . . . 11 (((𝑥 = 𝑏𝑦 = 𝑎) ∧ ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → ((𝑦𝑉𝑥𝑉) → 𝑦𝑅𝑥))
34 eleq1 2898 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → (𝑦𝑉𝑎𝑉))
35 eleq1 2898 . . . . . . . . . . . . . 14 (𝑥 = 𝑏 → (𝑥𝑉𝑏𝑉))
3634, 35bi2anan9r 638 . . . . . . . . . . . . 13 ((𝑥 = 𝑏𝑦 = 𝑎) → ((𝑦𝑉𝑥𝑉) ↔ (𝑎𝑉𝑏𝑉)))
37 breq12 5047 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑥 = 𝑏) → (𝑦𝑅𝑥𝑎𝑅𝑏))
3837ancoms 461 . . . . . . . . . . . . 13 ((𝑥 = 𝑏𝑦 = 𝑎) → (𝑦𝑅𝑥𝑎𝑅𝑏))
3936, 38imbi12d 347 . . . . . . . . . . . 12 ((𝑥 = 𝑏𝑦 = 𝑎) → (((𝑦𝑉𝑥𝑉) → 𝑦𝑅𝑥) ↔ ((𝑎𝑉𝑏𝑉) → 𝑎𝑅𝑏)))
4039adantr 483 . . . . . . . . . . 11 (((𝑥 = 𝑏𝑦 = 𝑎) ∧ ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → (((𝑦𝑉𝑥𝑉) → 𝑦𝑅𝑥) ↔ ((𝑎𝑉𝑏𝑉) → 𝑎𝑅𝑏)))
4133, 40mpbid 234 . . . . . . . . . 10 (((𝑥 = 𝑏𝑦 = 𝑎) ∧ ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦)) → ((𝑎𝑉𝑏𝑉) → 𝑎𝑅𝑏))
4241expimpd 456 . . . . . . . . 9 ((𝑥 = 𝑏𝑦 = 𝑎) → ((((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎𝑉𝑏𝑉)) → 𝑎𝑅𝑏))
4324, 42jaoi 853 . . . . . . . 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 244 . . . . . 6 ((((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) ∧ (𝑎𝑉𝑏𝑉)) → ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏))
4645ralrimivva 3178 . . . . 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 2902 . . . . . 6 ({𝑥, 𝑦} ∈ 𝑄 ↔ {𝑥, 𝑦} ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)})
49 eqeq1 2824 . . . . . . . . 9 (𝑞 = {𝑥, 𝑦} → (𝑞 = {𝑎, 𝑏} ↔ {𝑥, 𝑦} = {𝑎, 𝑏}))
5049imbi1d 344 . . . . . . . 8 (𝑞 = {𝑥, 𝑦} → ((𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
51502ralbidv 3186 . . . . . . 7 (𝑞 = {𝑥, 𝑦} → (∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ∀𝑎𝑉𝑏𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
5251elrab 3660 . . . . . 6 ({𝑥, 𝑦} ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ↔ ({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
5348, 52bitri 277 . . . . 5 ({𝑥, 𝑦} ∈ 𝑄 ↔ ({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 ({𝑥, 𝑦} = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
5413, 46, 53sylanbrc 585 . . . 4 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} ∈ 𝑄)
55 eqidd 2821 . . . 4 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → {𝑥, 𝑦} = {𝑥, 𝑦})
56 eqeq1 2824 . . . . 5 (𝑐 = {𝑥, 𝑦} → (𝑐 = {𝑥, 𝑦} ↔ {𝑥, 𝑦} = {𝑥, 𝑦}))
5756rspcev 3602 . . . 4 (({𝑥, 𝑦} ∈ 𝑄 ∧ {𝑥, 𝑦} = {𝑥, 𝑦}) → ∃𝑐𝑄 𝑐 = {𝑥, 𝑦})
5854, 55, 57syl2anc 586 . . 3 (((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦) → ∃𝑐𝑄 𝑐 = {𝑥, 𝑦})
5958ex 415 . 2 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 → ∃𝑐𝑄 𝑐 = {𝑥, 𝑦}))
6047eleq2i 2902 . . . . . 6 (𝑐𝑄𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)})
61 eqeq1 2824 . . . . . . . . 9 (𝑞 = 𝑐 → (𝑞 = {𝑎, 𝑏} ↔ 𝑐 = {𝑎, 𝑏}))
6261imbi1d 344 . . . . . . . 8 (𝑞 = 𝑐 → ((𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
63622ralbidv 3186 . . . . . . 7 (𝑞 = 𝑐 → (∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
6463elrab 3660 . . . . . 6 (𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ↔ (𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
6560, 64bitri 277 . . . . 5 (𝑐𝑄 ↔ (𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)))
66 eleq1 2898 . . . . . . . . . . 11 (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ (Pairs‘𝑉) ↔ {𝑥, 𝑦} ∈ (Pairs‘𝑉)))
67 prsprel 43795 . . . . . . . . . . . 12 (({𝑥, 𝑦} ∈ (Pairs‘𝑉) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝑉𝑦𝑉))
6814, 15, 67mpanr12 703 . . . . . . . . . . 11 ({𝑥, 𝑦} ∈ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))
6966, 68syl6bi 255 . . . . . . . . . 10 (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
7069com12 32 . . . . . . . . 9 (𝑐 ∈ (Pairs‘𝑉) → (𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
7170adantr 483 . . . . . . . 8 ((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) → (𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
7271imp 409 . . . . . . 7 (((𝑐 ∈ (Pairs‘𝑉) ∧ ∀𝑎𝑉𝑏𝑉 (𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏)) ∧ 𝑐 = {𝑥, 𝑦}) → (𝑥𝑉𝑦𝑉))
73 preq1 4645 . . . . . . . . . . . 12 (𝑎 = 𝑥 → {𝑎, 𝑏} = {𝑥, 𝑏})
7473eqeq2d 2831 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑥, 𝑏}))
75 breq1 5045 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑎𝑅𝑏𝑥𝑅𝑏))
7674, 75imbi12d 347 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑐 = {𝑎, 𝑏} → 𝑎𝑅𝑏) ↔ (𝑐 = {𝑥, 𝑏} → 𝑥𝑅𝑏)))
77 preq2 4646 . . . . . . . . . . . 12 (𝑏 = 𝑦 → {𝑥, 𝑏} = {𝑥, 𝑦})
7877eqeq2d 2831 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑐 = {𝑥, 𝑏} ↔ 𝑐 = {𝑥, 𝑦}))
79 breq2 5046 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑥𝑅𝑏𝑥𝑅𝑦))
8078, 79imbi12d 347 . . . . . . . . . 10 (𝑏 = 𝑦 → ((𝑐 = {𝑥, 𝑏} → 𝑥𝑅𝑏) ↔ (𝑐 = {𝑥, 𝑦} → 𝑥𝑅𝑦)))
8176, 80rspc2v 3612 . . . . . . . . 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 426 . . . . . . 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 583 . . . 4 ((𝑐𝑄𝑐 = {𝑥, 𝑦}) → ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → 𝑥𝑅𝑦))
8786rexlimiva 3268 . . 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 214 1 ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐𝑄 𝑐 = {𝑥, 𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843  w3a 1083   = wceq 1537  wcel 2114  wral 3125  wrex 3126  {crab 3129  Vcvv 3473  wss 3913  {cpr 4545  cop 4549   class class class wbr 5042   × cxp 5529  cfv 6331  Pairscspr 43785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-spr 43786
This theorem is referenced by:  sprsymrelfo  43805
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