Theorem elres 5049

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 5041	. . . . 5 ⊢ Rel (𝐵 ↾ 𝐶)
2		elrel 4828	. . . . 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 2294	. . . . . . . . 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 2805	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7	6	opelres 5018	. . . . . . . . . 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 563	. . . . . 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 1930	. . . 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 2826	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17		df-rex 2516	. . . . 5 ⊢ (∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
18	17	exbii 1653	. . . 4 ⊢ (∃𝑦∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
19		excom 1712	. . . 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 1489	. . . . . 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 1867	. . 3 ⊢ (𝑥 ∈ 𝐶 → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵 ↾ 𝐶)))
27	26	rexlimiv 2644	. 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 1397  ∃wex 1540   ∈ wcel 2202  ∃wrex 2511  ⟨cop 3672   ↾ cres 4727  Rel wrel 4730

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731  df-rel 4732  df-res 4737

This theorem is referenced by:  elsnres  5050