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Theorem sprsymrelfvlem 44424
 Description: Lemma for sprsymrelf 44429 and sprsymrelfv 44428. (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 486 . . . . 5 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑉 ∈ V)
2 eleq1 2839 . . . . . . . . . . . 12 (𝑐 = {𝑥, 𝑦} → (𝑐𝑃 ↔ {𝑥, 𝑦} ∈ 𝑃))
3 prsssprel 44422 . . . . . . . . . . . . . . 15 ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑥, 𝑦} ∈ 𝑃 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝑉𝑦𝑉))
433exp 1116 . . . . . . . . . . . . . 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 3417 . . . . . . . . . . . 12 ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
72, 6syl6bi 256 . . . . . . . . . . 11 (𝑐 = {𝑥, 𝑦} → (𝑐𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))))
87com12 32 . . . . . . . . . 10 (𝑐𝑃 → (𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉))))
98rexlimiv 3204 . . . . . . . . 9 (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
109com12 32 . . . . . . . 8 (𝑃 ⊆ (Pairs‘𝑉) → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
1110adantl 485 . . . . . . 7 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
1211imp 410 . . . . . 6 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → (𝑥𝑉𝑦𝑉))
1312simpld 498 . . . . 5 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → 𝑥𝑉)
1412simprd 499 . . . . 5 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → 𝑦𝑉)
151, 1, 13, 14opabex2 7765 . . . 4 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ V)
16 elopab 5388 . . . . . . 7 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}))
179adantl 485 . . . . . . . . . . . 12 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
1817adantld 494 . . . . . . . . . . 11 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑥𝑉𝑦𝑉)))
1918imp 410 . . . . . . . . . 10 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑥𝑉𝑦𝑉))
20 eleq1 2839 . . . . . . . . . . . 12 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
2120ad2antrr 725 . . . . . . . . . . 11 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
22 opelxp 5564 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉) ↔ (𝑥𝑉𝑦𝑉))
2321, 22bitrdi 290 . . . . . . . . . 10 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ (𝑥𝑉𝑦𝑉)))
2419, 23mpbird 260 . . . . . . . . 9 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → 𝑝 ∈ (𝑉 × 𝑉))
2524ex 416 . . . . . . . 8 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
2625exlimivv 1933 . . . . . . 7 (∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
2716, 26sylbi 220 . . . . . 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 3900 . . . 4 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ⊆ (𝑉 × 𝑉))
3015, 29elpwd 4505 . . 3 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3130ex 416 . 2 (𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
32 fvprc 6655 . . . . 5 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
3332sseq2d 3926 . . . 4 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 ⊆ ∅))
34 ss0b 4296 . . . 4 (𝑃 ⊆ ∅ ↔ 𝑃 = ∅)
3533, 34bitrdi 290 . . 3 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 = ∅))
36 rex0 4258 . . . . . . 7 ¬ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}
37 rexeq 3324 . . . . . . 7 (𝑃 = ∅ → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}))
3836, 37mtbiri 330 . . . . . 6 (𝑃 = ∅ → ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
3938alrimivv 1929 . . . . 5 (𝑃 = ∅ → ∀𝑥𝑦 ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
40 opab0 5415 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} = ∅ ↔ ∀𝑥𝑦 ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
4139, 40sylibr 237 . . . 4 (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} = ∅)
42 0elpw 5228 . . . 4 ∅ ∈ 𝒫 (𝑉 × 𝑉)
4341, 42eqeltrdi 2860 . . 3 (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
4435, 43syl6bi 256 . 2 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
4531, 44pm2.61i 185 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 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3071  Vcvv 3409   ⊆ wss 3860  ∅c0 4227  𝒫 cpw 4497  {cpr 4527  ⟨cop 4531  {copab 5098   × cxp 5526  ‘cfv 6340  Pairscspr 44411 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-spr 44412 This theorem is referenced by:  sprsymrelfv  44428  sprsymrelf  44429
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