Theorem 2elresin 5365

Description: Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)

Assertion

Ref	Expression
2elresin	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵)))))

Proof of Theorem 2elresin

Step	Hyp	Ref	Expression
1		fnop 5357	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥 ∈ 𝐴)
2		fnop 5357	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑥 ∈ 𝐵)
3	1, 2	anim12i 338	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
4	3	an4s 588	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
5		elin 3342	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
6	4, 5	sylibr 134	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → 𝑥 ∈ (𝐴 ∩ 𝐵))
7		vex 2763	. . . . . . . 8 ⊢ 𝑦 ∈ V
8	7	opres 4951	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9		vex 2763	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	9	opres 4951	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵)) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
11	8, 10	anbi12d 473	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵))) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)))
12	11	biimprd 158	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵)))))
13	6, 12	syl 14	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵)))))
14	13	ex 115	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵))))))
15	14	pm2.43d 50	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵)))))
16		resss 4966	. . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) ⊆ 𝐹
17	16	sseli 3175	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18		resss 4966	. . . 4 ⊢ (𝐺 ↾ (𝐴 ∩ 𝐵)) ⊆ 𝐺
19	18	sseli 3175	. . 3 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵)) → ⟨𝑥, 𝑧⟩ ∈ 𝐺)
20	17, 19	anim12i 338	. 2 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵))) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
21	15, 20	impbid1 142	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵)))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2164   ∩ cin 3152  ⟨cop 3621   ↾ cres 4661   Fn wfn 5249

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-dm 4669  df-res 4671  df-fun 5256  df-fn 5257

This theorem is referenced by: (None)