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Theorem 2elresin 4997
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 4989 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥𝐴)
2 fnop 4989 . . . . . . . 8 ((𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑥𝐵)
31, 2anim12i 321 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ∧ (𝐺 Fn 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥𝐴𝑥𝐵))
43an4s 522 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (𝑥𝐴𝑥𝐵))
5 elin 3123 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
64, 5sylibr 137 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → 𝑥 ∈ (𝐴𝐵))
7 vex 2557 . . . . . . . 8 𝑦 ∈ V
87opres 4608 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9 vex 2557 . . . . . . . 8 𝑧 ∈ V
109opres 4608 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
118, 10anbi12d 442 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵))) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)))
1211biimprd 147 . . . . 5 (𝑥 ∈ (𝐴𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)))))
136, 12syl 14 . . . 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 108 . . 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 44 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)))))
16 resss 4622 . . . 4 (𝐹 ↾ (𝐴𝐵)) ⊆ 𝐹
1716sseli 2938 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) → ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18 resss 4622 . . . 4 (𝐺 ↾ (𝐴𝐵)) ⊆ 𝐺
1918sseli 2938 . . 3 (⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)) → ⟨𝑥, 𝑧⟩ ∈ 𝐺)
2017, 19anim12i 321 . 2 ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵))) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
2115, 20impbid1 130 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wcel 1393  cin 2913  cop 3375  cres 4334   Fn wfn 4884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-br 3762  df-opab 3816  df-xp 4338  df-rel 4339  df-dm 4342  df-res 4344  df-fun 4891  df-fn 4892
This theorem is referenced by: (None)
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