Theorem sprsymrelfvlem 47477

Description: Lemma for sprsymrelf 47482 and sprsymrelfv 47481. (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 2829	. . . . . . . . . . . 12 ⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ 𝑃 ↔ {𝑥, 𝑦} ∈ 𝑃))
3		prsssprel 47475	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑥, 𝑦} ∈ 𝑃 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
4	3	3exp 1120	. . . . . . . . . . . . . 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 3487	. . . . . . . . . . . 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 3148	. . . . . . . . 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 8082	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ V)
16		elopab 5532	. . . . . . 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 2829	. . . . . . . . . . . 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 5721	. . . . . . . . . . 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 3989	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ⊆ (𝑉 × 𝑉))
30	15, 29	elpwd 4606	. . 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 6898	. . . . 5 ⊢ (¬ 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
33	32	sseq2d 4016	. . . 4 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 ⊆ ∅))
34		ss0b 4401	. . . 4 ⊢ (𝑃 ⊆ ∅ ↔ 𝑃 = ∅)
35	33, 34	bitrdi 287	. . 3 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 = ∅))
36		rex0 4360	. . . . . . 7 ⊢ ¬ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}
37		rexeq 3322	. . . . . . 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 5559	. . . . 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 5356	. . . 4 ⊢ ∅ ∈ 𝒫 (𝑉 × 𝑉)
43	41, 42	eqeltrdi 2849	. . 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 2108  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  {cpr 4628  ⟨cop 4632  {copab 5205   × cxp 5683  ‘cfv 6561  Pairscspr 47464

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-spr 47465

This theorem is referenced by:  sprsymrelfv  47481  sprsymrelf  47482