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Theorem sprsymrelfvlem 45772
Description: Lemma for sprsymrelf 45777 and sprsymrelfv 45776. (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 484 . . . . 5 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑉 ∈ V)
2 eleq1 2822 . . . . . . . . . . . 12 (𝑐 = {𝑥, 𝑦} → (𝑐𝑃 ↔ {𝑥, 𝑦} ∈ 𝑃))
3 prsssprel 45770 . . . . . . . . . . . . . . 15 ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑥, 𝑦} ∈ 𝑃 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝑉𝑦𝑉))
433exp 1120 . . . . . . . . . . . . . 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 3455 . . . . . . . . . . . 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 3142 . . . . . . . . 9 (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
109com12 32 . . . . . . . 8 (𝑃 ⊆ (Pairs‘𝑉) → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
1110adantl 483 . . . . . . 7 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} → (𝑥𝑉𝑦𝑉)))
1211imp 408 . . . . . 6 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → (𝑥𝑉𝑦𝑉))
1312simpld 496 . . . . 5 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → 𝑥𝑉)
1412simprd 497 . . . . 5 (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → 𝑦𝑉)
151, 1, 13, 14opabex2 7993 . . . 4 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ V)
16 elopab 5488 . . . . . . 7 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}))
179adantl 483 . . . . . . . . . . . 12 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥𝑉𝑦𝑉)))
1817adantld 492 . . . . . . . . . . 11 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑥𝑉𝑦𝑉)))
1918imp 408 . . . . . . . . . 10 (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑥𝑉𝑦𝑉))
20 eleq1 2822 . . . . . . . . . . . 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 5673 . . . . . . . . . . 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 414 . . . . . . . 8 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
2625exlimivv 1936 . . . . . . 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 3954 . . . 4 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ⊆ (𝑉 × 𝑉))
3015, 29elpwd 4570 . . 3 ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3130ex 414 . 2 (𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
32 fvprc 6838 . . . . 5 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
3332sseq2d 3980 . . . 4 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 ⊆ ∅))
34 ss0b 4361 . . . 4 (𝑃 ⊆ ∅ ↔ 𝑃 = ∅)
3533, 34bitrdi 287 . . 3 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 = ∅))
36 rex0 4321 . . . . . . 7 ¬ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}
37 rexeq 3309 . . . . . . 7 (𝑃 = ∅ → (∃𝑐𝑃 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}))
3836, 37mtbiri 327 . . . . . 6 (𝑃 = ∅ → ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
3938alrimivv 1932 . . . . 5 (𝑃 = ∅ → ∀𝑥𝑦 ¬ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦})
40 opab0 5515 . . . . 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 5315 . . . 4 ∅ ∈ 𝒫 (𝑉 × 𝑉)
4341, 42eqeltrdi 2842 . . 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 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  wrex 3070  Vcvv 3447  wss 3914  c0 4286  𝒫 cpw 4564  {cpr 4592  cop 4596  {copab 5171   × cxp 5635  cfv 6500  Pairscspr 45759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-spr 45760
This theorem is referenced by:  sprsymrelfv  45776  sprsymrelf  45777
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