Theorem sprsymrelfvlem 41740

Description: Lemma for sprsymrelf 41745 and sprsymrelfv 41744. (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 473	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑉 ∈ V)
2		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ 𝑃 ↔ {𝑥, 𝑦} ∈ 𝑃))
3		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
4		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
5		prsssprel 41738	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑥, 𝑦} ∈ 𝑃 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
6	5	3exp 1264	. . . . . . . . . . . . . 14 ⊢ (𝑃 ⊆ (Pairs‘𝑉) → ({𝑥, 𝑦} ∈ 𝑃 → ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))))
7	6	com13 88	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))))
8	3, 4, 7	mp2an 708	. . . . . . . . . . . 12 ⊢ ({𝑥, 𝑦} ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
9	2, 8	syl6bi 243	. . . . . . . . . . 11 ⊢ (𝑐 = {𝑥, 𝑦} → (𝑐 ∈ 𝑃 → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))))
10	9	com12 32	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝑃 → (𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))))
11	10	rexlimiv 3027	. . . . . . . . 9 ⊢ (∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦} → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
12	11	com12 32	. . . . . . . 8 ⊢ (𝑃 ⊆ (Pairs‘𝑉) → (∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦} → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
13	12	adantl 482	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦} → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
14	13	imp 445	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
15	14	simpld 475	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → 𝑥 ∈ 𝑉)
16	14	simprd 479	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → 𝑦 ∈ 𝑉)
17	1, 1, 15, 16	opabex2 7227	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ V)
18		elopab 4983	. . . . . . 7 ⊢ (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}))
19	11	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → (𝑃 ⊆ (Pairs‘𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
20	19	adantld 483	. . . . . . . . . . 11 ⊢ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
21	20	imp 445	. . . . . . . . . 10 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
22		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
23	22	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉)))
24		opelxp 5146	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑉 × 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
25	23, 24	syl6bb 276	. . . . . . . . . 10 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → (𝑝 ∈ (𝑉 × 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
26	21, 25	mpbird 247	. . . . . . . . 9 ⊢ (((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) ∧ (𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉))) → 𝑝 ∈ (𝑉 × 𝑉))
27	26	ex 450	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
28	27	exlimivv 1860	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}) → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
29	18, 28	sylbi 207	. . . . . 6 ⊢ (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} → ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → 𝑝 ∈ (𝑉 × 𝑉)))
30	29	com12 32	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} → 𝑝 ∈ (𝑉 × 𝑉)))
31	30	ssrdv 3609	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ⊆ (𝑉 × 𝑉))
32	17, 31	elpwd 4167	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝑃 ⊆ (Pairs‘𝑉)) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
33	32	ex 450	. 2 ⊢ (𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
34		fvprc 6185	. . . . 5 ⊢ (¬ 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
35	34	sseq2d 3633	. . . 4 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 ⊆ ∅))
36		ss0b 3973	. . . 4 ⊢ (𝑃 ⊆ ∅ ↔ 𝑃 = ∅)
37	35, 36	syl6bb 276	. . 3 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) ↔ 𝑃 = ∅))
38		rex0 3938	. . . . . . 7 ⊢ ¬ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}
39		rexeq 3139	. . . . . . 7 ⊢ (𝑃 = ∅ → (∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ ∅ 𝑐 = {𝑥, 𝑦}))
40	38, 39	mtbiri 317	. . . . . 6 ⊢ (𝑃 = ∅ → ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
41	40	alrimivv 1856	. . . . 5 ⊢ (𝑃 = ∅ → ∀𝑥∀𝑦 ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
42		opab0 5007	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} = ∅ ↔ ∀𝑥∀𝑦 ¬ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦})
43	41, 42	sylibr 224	. . . 4 ⊢ (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} = ∅)
44		0elpw 4834	. . . 4 ⊢ ∅ ∈ 𝒫 (𝑉 × 𝑉)
45	43, 44	syl6eqel 2709	. . 3 ⊢ (𝑃 = ∅ → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
46	37, 45	syl6bi 243	. 2 ⊢ (¬ 𝑉 ∈ V → (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)))
47	33, 46	pm2.61i 176	1 ⊢ (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {cpr 4179  ⟨cop 4183  {copab 4712   × cxp 5112  ‘cfv 5888  Pairscspr 41727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-spr 41728

This theorem is referenced by:  sprsymrelfv  41744  sprsymrelf  41745