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Theorem 2elresin 6451
 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 6442 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥𝐴)
2 fnop 6442 . . . . . . . 8 ((𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑥𝐵)
31, 2anim12i 615 . . . . . . 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 3874 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
64, 5sylibr 237 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → 𝑥 ∈ (𝐴𝐵))
7 vex 3413 . . . . . . . 8 𝑦 ∈ V
87opres 5833 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9 vex 3413 . . . . . . . 8 𝑧 ∈ V
109opres 5833 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
118, 10anbi12d 633 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵))) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)))
1211biimprd 251 . . . . 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 416 . . 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 5848 . . . 4 (𝐹 ↾ (𝐴𝐵)) ⊆ 𝐹
1716sseli 3888 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) → ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18 resss 5848 . . . 4 (𝐺 ↾ (𝐴𝐵)) ⊆ 𝐺
1918sseli 3888 . . 3 (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)) → ⟨𝑥, 𝑧⟩ ∈ 𝐺)
2017, 19anim12i 615 . 2 ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵))) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
2115, 20impbid1 228 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 209   ∧ wa 399   ∈ wcel 2111   ∩ cin 3857  ⟨cop 4528   ↾ cres 5526   Fn wfn 6330 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-xp 5530  df-rel 5531  df-dm 5534  df-res 5536  df-fun 6337  df-fn 6338 This theorem is referenced by: (None)
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