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Theorem 2elresin 6665
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 6652 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥𝐴)
2 fnop 6652 . . . . . . . 8 ((𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑥𝐵)
31, 2anim12i 612 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥𝐴𝑥𝐵))
43an4s 657 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥𝐴𝑥𝐵))
5 elin 3959 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
64, 5sylibr 233 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → 𝑥 ∈ (𝐴𝐵))
7 vex 3472 . . . . . . . 8 𝑦 ∈ V
87opres 5985 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9 vex 3472 . . . . . . . 8 𝑧 ∈ V
109opres 5985 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
118, 10anbi12d 630 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵))) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)))
1211biimprd 247 . . . . 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 412 . . 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 6000 . . . 4 (𝐹 ↾ (𝐴𝐵)) ⊆ 𝐹
1716sseli 3973 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) → ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18 resss 6000 . . . 4 (𝐺 ↾ (𝐴𝐵)) ⊆ 𝐺
1918sseli 3973 . . 3 (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)) → ⟨𝑥, 𝑧⟩ ∈ 𝐺)
2017, 19anim12i 612 . 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 395  wcel 2098  cin 3942  cop 4629  cres 5671   Fn wfn 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-dm 5679  df-res 5681  df-fun 6539  df-fn 6540
This theorem is referenced by: (None)
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