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Theorem sprsymrelfvlem 42242
Description: Lemma for sprsymrelf 42247 and sprsymrelfv 42246. (Contributed by AV, 19-Nov-2021.)
Assertion
Ref Expression
sprsymrelfvlem (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
Distinct variable groups:   𝑃,𝑐,𝑥,𝑦   𝑉,𝑐,𝑥,𝑦

Proof of Theorem sprsymrelfvlem
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simpl 474 . . . . 5 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑉 ∈ V)
2 eleq1 2819 . . . . . . . . . . . 12 (𝑐 = {𝑥, 𝑦} → (𝑐𝑃 ↔ {𝑥, 𝑦} ∈ 𝑃))
3 vex 3335 . . . . . . . . . . . . 13 𝑥 ∈ V
4 vex 3335 . . . . . . . . . . . . 13 𝑦 ∈ V
5 prsssprel 42240 . . . . . . . . . . . . . . 15 ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑥, 𝑦} ∈ 𝑃 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝑉𝑦𝑉))
653exp 1112 . . . . . . . . . . . . . 14 (𝑃 ⊆ (Pairs‘𝑉) → ({𝑥, 𝑦} ∈ 𝑃 → ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑉𝑦𝑉))))
76com13 88 . . . . . . . . . . . . 13 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))))
83, 4, 7mp2an 710 . . . . . . . . . . . 12 ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
92, 8syl6bi 243 . . . . . . . . . . 11 (𝑐 = {𝑥, 𝑦} → (𝑐𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))))
109com12 32 . . . . . . . . . 10 (𝑐𝑃 → (𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))))
1110rexlimiv 3157 . . . . . . . . 9 (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
1211com12 32 . . . . . . . 8 (𝑃 ⊆ (Pairs‘𝑉) → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
1312adantl 473 . . . . . . 7 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
1413imp 444 . . . . . 6 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → (𝑥𝑉𝑦𝑉))
1514simpld 477 . . . . 5 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → 𝑥𝑉)
1614simprd 482 . . . . 5 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → 𝑦𝑉)
171, 1, 15, 16opabex2 7386 . . . 4 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ V)
18 elopab 5125 . . . . . . 7 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}))
1911adantl 473 . . . . . . . . . . . 12 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
2019adantld 484 . . . . . . . . . . 11 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑥𝑉𝑦𝑉)))
2120imp 444 . . . . . . . . . 10 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑥𝑉𝑦𝑉))
22 eleq1 2819 . . . . . . . . . . . 12 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
2322ad2antrr 764 . . . . . . . . . . 11 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
24 opelxp 5295 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉) ↔ (𝑥𝑉𝑦𝑉))
2523, 24syl6bb 276 . . . . . . . . . 10 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ (𝑥𝑉𝑦𝑉)))
2621, 25mpbird 247 . . . . . . . . 9 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → 𝑝 ∈ (𝑉 × 𝑉))
2726ex 449 . . . . . . . 8 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
2827exlimivv 2001 . . . . . . 7 (∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
2918, 28sylbi 207 . . . . . 6 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
3029com12 32 . . . . 5 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} → 𝑝 ∈ (𝑉 × 𝑉)))
3130ssrdv 3742 . . . 4 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ⊆ (𝑉 × 𝑉))
3217, 31elpwd 4303 . . 3 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3332ex 449 . 2 (𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
34 fvprc 6338 . . . . 5 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
3534sseq2d 3766 . . . 4 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 ⊆ ∅))
36 ss0b 4108 . . . 4 (𝑃 ⊆ ∅ ↔ 𝑃 = ∅)
3735, 36syl6bb 276 . . 3 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 = ∅))
38 rex0 4073 . . . . . . 7 ¬ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}
39 rexeq 3270 . . . . . . 7 (𝑃 = ∅ → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}))
4038, 39mtbiri 316 . . . . . 6 (𝑃 = ∅ → ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
4140alrimivv 1997 . . . . 5 (𝑃 = ∅ → ∀𝑥𝑦 ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
42 opab0 5149 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} = ∅ ↔ ∀𝑥𝑦 ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
4341, 42sylibr 224 . . . 4 (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} = ∅)
44 0elpw 4975 . . . 4 ∅ ∈ 𝒫 (𝑉 × 𝑉)
4543, 44syl6eqel 2839 . . 3 (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
4637, 45syl6bi 243 . 2 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
4733, 46pm2.61i 176 1 (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1622   = wceq 1624  wex 1845  wcel 2131  wrex 3043  Vcvv 3332  wss 3707  c0 4050  𝒫 cpw 4294  {cpr 4315  cop 4319  {copab 4856   × cxp 5256  cfv 6041  Pairscspr 42229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-spr 42230
This theorem is referenced by:  sprsymrelfv  42246  sprsymrelf  42247
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