Theorem 2elresin 6655

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 6643	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥 ∈ 𝐴)
2		fnop 6643	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑥 ∈ 𝐵)
3	1, 2	anim12i 613	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
4	3	an4s 660	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
5		elin 3940	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
6	4, 5	sylibr 234	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → 𝑥 ∈ (𝐴 ∩ 𝐵))
7		vex 3461	. . . . . . . 8 ⊢ 𝑦 ∈ V
8	7	opres 5973	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9		vex 3461	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	9	opres 5973	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵)) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
11	8, 10	anbi12d 632	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵))) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)))
12	11	biimprd 248	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵)))))
13	6, 12	syl 17	. . . 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 412	. . 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 53	. 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 5985	. . . 4 ⊢ (𝐹 ↾ (𝐴 ∩ 𝐵)) ⊆ 𝐹
17	16	sseli 3952	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18		resss 5985	. . . 4 ⊢ (𝐺 ↾ (𝐴 ∩ 𝐵)) ⊆ 𝐺
19	18	sseli 3952	. . 3 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵)) → ⟨𝑥, 𝑧⟩ ∈ 𝐺)
20	17, 19	anim12i 613	. 2 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵))) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
21	15, 20	impbid1 225	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   ↔ wb 206   ∧ wa 395   ∈ wcel 2107   ∩ cin 3923  ⟨cop 4605   ↾ cres 5653   Fn wfn 6522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pr 5399

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-br 5117  df-opab 5179  df-xp 5657  df-rel 5658  df-dm 5661  df-res 5663  df-fun 6529  df-fn 6530

This theorem is referenced by: (None)