MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2elresin Structured version   Visualization version   GIF version

Theorem 2elresin 6623
Description: Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
Assertion
Ref Expression
2elresin ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)))))

Proof of Theorem 2elresin
StepHypRef Expression
1 fnop 6612 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥𝐴)
2 fnop 6612 . . . . . . . 8 ((𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑥𝐵)
31, 2anim12i 614 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥𝐴𝑥𝐵))
43an4s 659 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥𝐴𝑥𝐵))
5 elin 3927 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
64, 5sylibr 233 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → 𝑥 ∈ (𝐴𝐵))
7 vex 3448 . . . . . . . 8 𝑦 ∈ V
87opres 5948 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9 vex 3448 . . . . . . . 8 𝑧 ∈ V
109opres 5948 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
118, 10anbi12d 632 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵))) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)))
1211biimprd 248 . . . . 5 (𝑥 ∈ (𝐴𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)))))
136, 12syl 17 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)))))
1413ex 414 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵))))))
1514pm2.43d 53 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)))))
16 resss 5963 . . . 4 (𝐹 ↾ (𝐴𝐵)) ⊆ 𝐹
1716sseli 3941 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) → ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18 resss 5963 . . . 4 (𝐺 ↾ (𝐴𝐵)) ⊆ 𝐺
1918sseli 3941 . . 3 (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)) → ⟨𝑥, 𝑧⟩ ∈ 𝐺)
2017, 19anim12i 614 . 2 ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵))) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
2115, 20impbid1 224 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  cin 3910  cop 4593  cres 5636   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-dm 5644  df-res 5646  df-fun 6499  df-fn 6500
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator