Theorem sprsymrelfvlem 47495

Description: Lemma for sprsymrelf 47500 and sprsymrelfv 47499. (Contributed by AV, 19-Nov-2021.)

Assertion

Ref	Expression
sprsymrelfvlem	⊢ (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 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.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑉 ∈ V)
2		eleq1 2817	. . . . . . . . . . . 12 ⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ 𝑃 ↔ {𝑥, 𝑦} ∈ 𝑃))
3		prsssprel 47493	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑥, 𝑦} ∈ 𝑃 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
4	3	3exp 1119	. . . . . . . . . . . . . 14 ⊢ (𝑃 ⊆ (Pairs‘𝑉) → ({𝑥, 𝑦} ∈ 𝑃 → ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))))
5	4	com13 88	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))))
6	5	el2v 3457	. . . . . . . . . . . 12 ⊢ ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
7	2, 6	biimtrdi 253	. . . . . . . . . . 11 ⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))))
8	7	com12 32	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝑃 → (𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))))
9	8	rexlimiv 3128	. . . . . . . . 9 ⊢ (∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
10	9	com12 32	. . . . . . . 8 ⊢ (𝑃 ⊆ (Pairs‘𝑉) → (∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦} → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
11	10	adantl 481	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦} → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
12	11	imp 406	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
13	12	simpld 494	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → 𝑥 ∈ 𝑉)
14	12	simprd 495	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → 𝑦 ∈ 𝑉)
15	1, 1, 13, 14	opabex2 8039	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ V)
16		elopab 5490	. . . . . . 7 ⊢ (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}))
17	9	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
18	17	adantld 490	. . . . . . . . . . 11 ⊢ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
19	18	imp 406	. . . . . . . . . 10 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
20		eleq1 2817	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
21	20	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
22		opelxp 5677	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
23	21, 22	bitrdi 287	. . . . . . . . . 10 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
24	19, 23	mpbird 257	. . . . . . . . 9 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → 𝑝 ∈ (𝑉 × 𝑉))
25	24	ex 412	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
26	25	exlimivv 1932	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
27	16, 26	sylbi 217	. . . . . 6 ⊢ (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
28	27	com12 32	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} → 𝑝 ∈ (𝑉 × 𝑉)))
29	28	ssrdv 3955	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ⊆ (𝑉 × 𝑉))
30	15, 29	elpwd 4572	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
31	30	ex 412	. 2 ⊢ (𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
32		fvprc 6853	. . . . 5 ⊢ (¬ 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
33	32	sseq2d 3982	. . . 4 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 ⊆ ∅))
34		ss0b 4367	. . . 4 ⊢ (𝑃 ⊆ ∅ ↔ 𝑃 = ∅)
35	33, 34	bitrdi 287	. . 3 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 = ∅))
36		rex0 4326	. . . . . . 7 ⊢ ¬ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}
37		rexeq 3297	. . . . . . 7 ⊢ (𝑃 = ∅ → (∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}))
38	36, 37	mtbiri 327	. . . . . 6 ⊢ (𝑃 = ∅ → ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
39	38	alrimivv 1928	. . . . 5 ⊢ (𝑃 = ∅ → ∀𝑥∀𝑦 ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
40		opab0 5517	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} = ∅ ↔ ∀𝑥∀𝑦 ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
41	39, 40	sylibr 234	. . . 4 ⊢ (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} = ∅)
42		0elpw 5314	. . . 4 ⊢ ∅ ∈ 𝒫 (𝑉 × 𝑉)
43	41, 42	eqeltrdi 2837	. . 3 ⊢ (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
44	35, 43	biimtrdi 253	. 2 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
45	31, 44	pm2.61i 182	1 ⊢ (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {cpr 4594  ⟨cop 4598  {copab 5172   × cxp 5639  ‘cfv 6514  Pairscspr 47482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-spr 47483

This theorem is referenced by:  sprsymrelfv  47499  sprsymrelf  47500