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⟩ ∣ ∃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 486	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑉 ∈ V)
2		eleq1 2839	. . . . . . . . . . . 12 ⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ 𝑃 ↔ {𝑥, 𝑦} ∈ 𝑃))
3		prsssprel 44422	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑥, 𝑦} ∈ 𝑃 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
4	3	3exp 1116	. . . . . . . . . . . . . 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 3417	. . . . . . . . . . . 12 ⊢ ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
7	2, 6	syl6bi 256	. . . . . . . . . . 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 3204	. . . . . . . . 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 485	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦} → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
12	11	imp 410	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
13	12	simpld 498	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → 𝑥 ∈ 𝑉)
14	12	simprd 499	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → 𝑦 ∈ 𝑉)
15	1, 1, 13, 14	opabex2 7765	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ V)
16		elopab 5388	. . . . . . 7 ⊢ (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}))
17	9	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
18	17	adantld 494	. . . . . . . . . . 11 ⊢ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
19	18	imp 410	. . . . . . . . . 10 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
20		eleq1 2839	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
21	20	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
22		opelxp 5564	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
23	21, 22	bitrdi 290	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → 𝑝 ∈ (𝑉 × 𝑉))
25	24	ex 416	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
26	25	exlimivv 1933	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
27	16, 26	sylbi 220	. . . . . 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 3900	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ⊆ (𝑉 × 𝑉))
30	15, 29	elpwd 4505	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
31	30	ex 416	. 2 ⊢ (𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
32		fvprc 6655	. . . . 5 ⊢ (¬ 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
33	32	sseq2d 3926	. . . 4 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 ⊆ ∅))
34		ss0b 4296	. . . 4 ⊢ (𝑃 ⊆ ∅ ↔ 𝑃 = ∅)
35	33, 34	bitrdi 290	. . 3 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 = ∅))
36		rex0 4258	. . . . . . 7 ⊢ ¬ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}
37		rexeq 3324	. . . . . . 7 ⊢ (𝑃 = ∅ → (∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}))
38	36, 37	mtbiri 330	. . . . . 6 ⊢ (𝑃 = ∅ → ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
39	38	alrimivv 1929	. . . . 5 ⊢ (𝑃 = ∅ → ∀𝑥∀𝑦 ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
40		opab0 5415	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} = ∅ ↔ ∀𝑥∀𝑦 ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
41	39, 40	sylibr 237	. . . 4 ⊢ (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} = ∅)
42		0elpw 5228	. . . 4 ⊢ ∅ ∈ 𝒫 (𝑉 × 𝑉)
43	41, 42	eqeltrdi 2860	. . 3 ⊢ (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
44	35, 43	syl6bi 256	. 2 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
45	31, 44	pm2.61i 185	1 ⊢ (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))

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