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Theorem sprsymrelfvlem 43503
Description: Lemma for sprsymrelf 43508 and sprsymrelfv 43507. (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 483 . . . . 5 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑉 ∈ V)
2 eleq1 2905 . . . . . . . . . . . 12 (𝑐 = {𝑥, 𝑦} → (𝑐𝑃 ↔ {𝑥, 𝑦} ∈ 𝑃))
3 prsssprel 43501 . . . . . . . . . . . . . . 15 ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑥, 𝑦} ∈ 𝑃 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝑉𝑦𝑉))
433exp 1113 . . . . . . . . . . . . . 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 3507 . . . . . . . . . . . 12 ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
72, 6syl6bi 254 . . . . . . . . . . 11 (𝑐 = {𝑥, 𝑦} → (𝑐𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))))
87com12 32 . . . . . . . . . 10 (𝑐𝑃 → (𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))))
98rexlimiv 3285 . . . . . . . . 9 (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
109com12 32 . . . . . . . 8 (𝑃 ⊆ (Pairs‘𝑉) → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
1110adantl 482 . . . . . . 7 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
1211imp 407 . . . . . 6 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → (𝑥𝑉𝑦𝑉))
1312simpld 495 . . . . 5 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → 𝑥𝑉)
1412simprd 496 . . . . 5 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → 𝑦𝑉)
151, 1, 13, 14opabex2 7751 . . . 4 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ V)
16 elopab 5411 . . . . . . 7 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}))
179adantl 482 . . . . . . . . . . . 12 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
1817adantld 491 . . . . . . . . . . 11 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑥𝑉𝑦𝑉)))
1918imp 407 . . . . . . . . . 10 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑥𝑉𝑦𝑉))
20 eleq1 2905 . . . . . . . . . . . 12 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
2120ad2antrr 722 . . . . . . . . . . 11 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
22 opelxp 5590 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉) ↔ (𝑥𝑉𝑦𝑉))
2321, 22syl6bb 288 . . . . . . . . . 10 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ (𝑥𝑉𝑦𝑉)))
2419, 23mpbird 258 . . . . . . . . 9 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → 𝑝 ∈ (𝑉 × 𝑉))
2524ex 413 . . . . . . . 8 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
2625exlimivv 1926 . . . . . . 7 (∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
2716, 26sylbi 218 . . . . . 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 3977 . . . 4 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ⊆ (𝑉 × 𝑉))
3015, 29elpwd 4553 . . 3 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3130ex 413 . 2 (𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
32 fvprc 6662 . . . . 5 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
3332sseq2d 4003 . . . 4 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 ⊆ ∅))
34 ss0b 4355 . . . 4 (𝑃 ⊆ ∅ ↔ 𝑃 = ∅)
3533, 34syl6bb 288 . . 3 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 = ∅))
36 rex0 4321 . . . . . . 7 ¬ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}
37 rexeq 3412 . . . . . . 7 (𝑃 = ∅ → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}))
3836, 37mtbiri 328 . . . . . 6 (𝑃 = ∅ → ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
3938alrimivv 1922 . . . . 5 (𝑃 = ∅ → ∀𝑥𝑦 ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
40 opab0 5438 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} = ∅ ↔ ∀𝑥𝑦 ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
4139, 40sylibr 235 . . . 4 (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} = ∅)
42 0elpw 5253 . . . 4 ∅ ∈ 𝒫 (𝑉 × 𝑉)
4341, 42syl6eqel 2926 . . 3 (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
4435, 43syl6bi 254 . 2 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
4531, 44pm2.61i 183 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 207  wa 396  wal 1528   = wceq 1530  wex 1773  wcel 2107  wrex 3144  Vcvv 3500  wss 3940  c0 4295  𝒫 cpw 4542  {cpr 4566  cop 4570  {copab 5125   × cxp 5552  cfv 6354  Pairscspr 43490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-spr 43491
This theorem is referenced by:  sprsymrelfv  43507  sprsymrelf  43508
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