Theorem sprsymrelfvlem 48057

Description: Lemma for sprsymrelf 48062 and sprsymrelfv 48061. (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 2849	. . . . . . . . . . . 12 ⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ 𝑃 ↔ {𝑥, 𝑦} ∈ 𝑃))
3		prsssprel 48055	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑥, 𝑦} ∈ 𝑃 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
4	3	3exp 1131	. . . . . . . . . . . . . 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 3460	. . . . . . . . . . . 12 ⊢ ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
7	2, 6	biimtrdi 255	. . . . . . . . . . 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 3155	. . . . . . . . 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 8033	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ V)
16		elopab 5494	. . . . . . 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 2849	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
21	20	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
22		opelxp 5679	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
23	21, 22	bitrdi 289	. . . . . . . . . 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 259	. . . . . . . . 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 1951	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
27	16, 26	sylbi 219	. . . . . 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 3940	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ⊆ (𝑉 × 𝑉))
30	15, 29	elpwd 4558	. . 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 6854	. . . . 5 ⊢ (¬ 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
33	32	sseq2d 3966	. . . 4 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 ⊆ ∅))
34		ss0b 4352	. . . 4 ⊢ (𝑃 ⊆ ∅ ↔ 𝑃 = ∅)
35	33, 34	bitrdi 289	. . 3 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 = ∅))
36		rex0 4310	. . . . . . 7 ⊢ ¬ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}
37		rexeq 3315	. . . . . . 7 ⊢ (𝑃 = ∅ → (∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}))
38	36, 37	mtbiri 329	. . . . . 6 ⊢ (𝑃 = ∅ → ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
39	38	alrimivv 1947	. . . . 5 ⊢ (𝑃 = ∅ → ∀𝑥∀𝑦 ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
40		opab0 5521	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} = ∅ ↔ ∀𝑥∀𝑦 ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
41	39, 40	sylibr 236	. . . 4 ⊢ (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} = ∅)
42		0elpw 5309	. . . 4 ⊢ ∅ ∈ 𝒫 (𝑉 × 𝑉)
43	41, 42	eqeltrdi 2869	. . 3 ⊢ (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
44	35, 43	biimtrdi 255	. 2 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
45	31, 44	pm2.61i 183	1 ⊢ (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∃wrex 3085  Vcvv 3453   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4552  {cpr 4581  ⟨cop 4585  {copab 5159   × cxp 5641  ‘cfv 6516  Pairscspr 48044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524  df-spr 48045

This theorem is referenced by:  sprsymrelfv  48061  sprsymrelf  48062