Theorem elres 5040

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 5032	. . . . 5 ⊢ Rel (𝐵 ↾ 𝐶)
2		elrel 4820	. . . . 5 ⊢ ((Rel (𝐵 ↾ 𝐶) ∧ 𝐴 ∈ (𝐵 ↾ 𝐶)) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
3	1, 2	mpan 424	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
4		eleq1 2292	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶)))
5	4	biimpd 144	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ (𝐵 ↾ 𝐶) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶)))
6		vex 2802	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7	6	opelres 5009	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
8	7	biimpi 120	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶) → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
9	8	ancomd 267	. . . . . . . 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 325	. . . . . 6 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
12		an12 561	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) ↔ (𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
13	11, 12	imbitrdi 161	. . . . 5 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))))
14	13	2eximdv 1928	. . . 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 2823	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17		df-rex 2514	. . . . 5 ⊢ (∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
18	17	exbii 1651	. . . 4 ⊢ (∃𝑦∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
19		excom 1710	. . . 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 206	. . 3 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
21	15, 20	sylibr 134	. 2 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
22	7	simplbi2com 1487	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵 ↾ 𝐶)))
23	4	biimprd 158	. . . . . 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 254	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵 ↾ 𝐶)))
26	25	exlimdv 1865	. . 3 ⊢ (𝑥 ∈ 𝐶 → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵 ↾ 𝐶)))
27	26	rexlimiv 2642	. 2 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵 ↾ 𝐶))
28	21, 27	impbii 126	1 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509  ⟨cop 3669   ↾ cres 4720  Rel wrel 4723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4145  df-xp 4724  df-rel 4725  df-res 4730

This theorem is referenced by:  elsnres  5041