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Theorem sprsymrelfvlem 46457
Description: Lemma for sprsymrelf 46462 and sprsymrelfv 46461. (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 482 . . . . 5 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑉 ∈ V)
2 eleq1 2820 . . . . . . . . . . . 12 (𝑐 = {𝑥, 𝑦} → (𝑐𝑃 ↔ {𝑥, 𝑦} ∈ 𝑃))
3 prsssprel 46455 . . . . . . . . . . . . . . 15 ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑥, 𝑦} ∈ 𝑃 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝑉𝑦𝑉))
433exp 1118 . . . . . . . . . . . . . 14 (𝑃 ⊆ (Pairs‘𝑉) → ({𝑥, 𝑦} ∈ 𝑃 → ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑉𝑦𝑉))))
54com13 88 . . . . . . . . . . . . 13 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))))
65el2v 3481 . . . . . . . . . . . 12 ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
72, 6syl6bi 253 . . . . . . . . . . 11 (𝑐 = {𝑥, 𝑦} → (𝑐𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))))
87com12 32 . . . . . . . . . 10 (𝑐𝑃 → (𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))))
98rexlimiv 3147 . . . . . . . . 9 (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
109com12 32 . . . . . . . 8 (𝑃 ⊆ (Pairs‘𝑉) → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
1110adantl 481 . . . . . . 7 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
1211imp 406 . . . . . 6 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → (𝑥𝑉𝑦𝑉))
1312simpld 494 . . . . 5 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → 𝑥𝑉)
1412simprd 495 . . . . 5 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → 𝑦𝑉)
151, 1, 13, 14opabex2 8047 . . . 4 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ V)
16 elopab 5527 . . . . . . 7 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}))
179adantl 481 . . . . . . . . . . . 12 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
1817adantld 490 . . . . . . . . . . 11 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑥𝑉𝑦𝑉)))
1918imp 406 . . . . . . . . . 10 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑥𝑉𝑦𝑉))
20 eleq1 2820 . . . . . . . . . . . 12 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
2120ad2antrr 723 . . . . . . . . . . 11 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
22 opelxp 5712 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉) ↔ (𝑥𝑉𝑦𝑉))
2321, 22bitrdi 287 . . . . . . . . . 10 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ (𝑥𝑉𝑦𝑉)))
2419, 23mpbird 257 . . . . . . . . 9 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → 𝑝 ∈ (𝑉 × 𝑉))
2524ex 412 . . . . . . . 8 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
2625exlimivv 1934 . . . . . . 7 (∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
2716, 26sylbi 216 . . . . . 6 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
2827com12 32 . . . . 5 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} → 𝑝 ∈ (𝑉 × 𝑉)))
2928ssrdv 3988 . . . 4 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ⊆ (𝑉 × 𝑉))
3015, 29elpwd 4608 . . 3 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3130ex 412 . 2 (𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
32 fvprc 6883 . . . . 5 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
3332sseq2d 4014 . . . 4 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 ⊆ ∅))
34 ss0b 4397 . . . 4 (𝑃 ⊆ ∅ ↔ 𝑃 = ∅)
3533, 34bitrdi 287 . . 3 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 = ∅))
36 rex0 4357 . . . . . . 7 ¬ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}
37 rexeq 3320 . . . . . . 7 (𝑃 = ∅ → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}))
3836, 37mtbiri 327 . . . . . 6 (𝑃 = ∅ → ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
3938alrimivv 1930 . . . . 5 (𝑃 = ∅ → ∀𝑥𝑦 ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
40 opab0 5554 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} = ∅ ↔ ∀𝑥𝑦 ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
4139, 40sylibr 233 . . . 4 (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} = ∅)
42 0elpw 5354 . . . 4 ∅ ∈ 𝒫 (𝑉 × 𝑉)
4341, 42eqeltrdi 2840 . . 3 (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
4435, 43syl6bi 253 . 2 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
4531, 44pm2.61i 182 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 205  wa 395  wal 1538   = wceq 1540  wex 1780  wcel 2105  wrex 3069  Vcvv 3473  wss 3948  c0 4322  𝒫 cpw 4602  {cpr 4630  cop 4634  {copab 5210   × cxp 5674  cfv 6543  Pairscspr 46444
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-spr 46445
This theorem is referenced by:  sprsymrelfv  46461  sprsymrelf  46462
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