Theorem elres 5094

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 5086	. . . . 5 ⊢ Rel (𝐵 ↾ 𝐶)
2		elrel 4872	. . . . 5 ⊢ ((Rel (𝐵 ↾ 𝐶) ∧ 𝐴 ∈ (𝐵 ↾ 𝐶)) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
3	1, 2	mpan 428	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
4		eleq1 2301	. . . . . . . . 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 2824	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7	6	opelres 5063	. . . . . . . . . 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 567	. . . . . 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 1935	. . . 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 2845	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17		df-rex 2534	. . . . 5 ⊢ (∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
18	17	exbii 1658	. . . 4 ⊢ (∃𝑦∃𝑥 ∈ 𝐶 (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥(𝑥 ∈ 𝐶 ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
19		excom 1716	. . . 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 1494	. . . . . 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 1872	. . 3 ⊢ (𝑥 ∈ 𝐶 → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ∈ (𝐵 ↾ 𝐶)))
27	26	rexlimiv 2662	. 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 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529  ⟨cop 3708   ↾ cres 4771  Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188  df-xp 4775  df-rel 4776  df-res 4781

This theorem is referenced by:  elsnres  5095