Theorem elres 4920

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 4912	. . . . 5 ⊢ Rel (𝐵 ↾ 𝐶)
2		elrel 4706	. . . . 5 ⊢ ((Rel (𝐵 ↾ 𝐶) ∧ 𝐴 ∈ (𝐵 ↾ 𝐶)) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
3	1, 2	mpan 421	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
4		eleq1 2229	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶)))
5	4	biimpd 143	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵 ↾ 𝐶) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶)))
6		vex 2729	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7	6	opelres 4889	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
8	7	biimpi 119	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶) → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
9	8	ancomd 265	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶) → (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
10	5, 9	syl6com 35	. . . . . . 7 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
11	10	ancld 323	. . . . . 6 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
12		an12 551	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
13	11, 12	syl6ib 160	. . . . 5 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
14	13	2eximdv 1870	. . . 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 2749	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17		df-rex 2450	. . . . 5 ⊢ (∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
18	17	exbii 1593	. . . 4 ⊢ (∃𝑦∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
19		excom 1652	. . . 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 205	. . 3 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
21	15, 20	sylibr 133	. 2 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
22	7	simplbi2com 1432	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶)))
23	4	biimprd 157	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶) → 𝐴 ∈ (𝐵 ↾ 𝐶)))
24	22, 23	syl9 72	. . . . 5 ⊢ (𝑥 ∈ 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝐵 → 𝐴 ∈ (𝐵 ↾ 𝐶))))
25	24	impd 252	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵 ↾ 𝐶)))
26	25	exlimdv 1807	. . 3 ⊢ (𝑥 ∈ 𝐶 → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵 ↾ 𝐶)))
27	26	rexlimiv 2577	. 2 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵 ↾ 𝐶))
28	21, 27	impbii 125	1 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∃wrex 2445  ⟨cop 3579   ↾ cres 4606  Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-xp 4610  df-rel 4611  df-res 4616

This theorem is referenced by:  elsnres  4921