Theorem elres 4646

Description: Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)

Assertion

Ref	Expression
elres	⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem elres

Step	Hyp	Ref	Expression
1		relres 4639	. . . . 5 ⊢ Rel (𝐵 ↾ 𝐶)
2		elrel 4442	. . . . 5 ⊢ ((Rel (𝐵 ↾ 𝐶) ∧ 𝐴 ∈ (𝐵 ↾ 𝐶)) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
3	1, 2	mpan 400	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
4		eleq1 2100	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶)))
5	4	biimpd 132	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵 ↾ 𝐶) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶)))
6		vex 2560	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7	6	opelres 4617	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
8	7	biimpi 113	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶) → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
9	8	ancomd 254	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶) → (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
10	5, 9	syl6com 31	. . . . . . 7 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
11	10	ancld 308	. . . . . 6 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
12		an12 495	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
13	11, 12	syl6ib 150	. . . . 5 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
14	13	2eximdv 1762	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
15	3, 14	mpd 13	. . 3 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → ∃𝑥∃𝑦(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
16		rexcom4 2577	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17		df-rex 2312	. . . . 5 ⊢ (∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
18	17	exbii 1496	. . . 4 ⊢ (∃𝑦∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
19		excom 1554	. . . 4 ⊢ (∃𝑦∃𝑥(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
20	16, 18, 19	3bitri 195	. . 3 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
21	15, 20	sylibr 137	. 2 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
22	7	simplbi2com 1333	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶)))
23	4	biimprd 147	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶) → 𝐴 ∈ (𝐵 ↾ 𝐶)))
24	22, 23	syl9 66	. . . . 5 ⊢ (𝑥 ∈ 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝐵 → 𝐴 ∈ (𝐵 ↾ 𝐶))))
25	24	impd 242	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵 ↾ 𝐶)))
26	25	exlimdv 1700	. . 3 ⊢ (𝑥 ∈ 𝐶 → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵 ↾ 𝐶)))
27	26	rexlimiv 2427	. 2 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵 ↾ 𝐶))
28	21, 27	impbii 117	1 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378   ↾ cres 4347  Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352  df-res 4357

This theorem is referenced by:  elsnres  4647