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Theorem 2elresin 6142
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 6134 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥𝐴)
2 fnop 6134 . . . . . . . 8 ((𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑥𝐵)
31, 2anim12i 592 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥𝐴𝑥𝐵))
43an4s 631 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥𝐴𝑥𝐵))
5 elin 3945 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
64, 5sylibr 224 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → 𝑥 ∈ (𝐴𝐵))
7 vex 3352 . . . . . . . 8 𝑦 ∈ V
87opres 5547 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9 vex 3352 . . . . . . . 8 𝑧 ∈ V
109opres 5547 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
118, 10anbi12d 608 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵))) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)))
1211biimprd 238 . . . . 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 397 . . 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 5563 . . . 4 (𝐹 ↾ (𝐴𝐵)) ⊆ 𝐹
1716sseli 3746 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) → ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18 resss 5563 . . . 4 (𝐺 ↾ (𝐴𝐵)) ⊆ 𝐺
1918sseli 3746 . . 3 (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)) → ⟨𝑥, 𝑧⟩ ∈ 𝐺)
2017, 19anim12i 592 . 2 ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵))) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
2115, 20impbid1 215 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 196  wa 382  wcel 2144  cin 3720  cop 4320  cres 5251   Fn wfn 6026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-dm 5259  df-res 5261  df-fun 6033  df-fn 6034
This theorem is referenced by: (None)
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