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Theorem opelf 5369
Description: The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opelf ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶𝐴𝐷𝐵))

Proof of Theorem opelf
StepHypRef Expression
1 fssxp 5365 . . . 4 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
21sseld 3146 . . 3 (𝐹:𝐴𝐵 → (⟨𝐶, 𝐷⟩ ∈ 𝐹 → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)))
3 opelxp 4641 . . 3 (⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵) ↔ (𝐶𝐴𝐷𝐵))
42, 3syl6ib 160 . 2 (𝐹:𝐴𝐵 → (⟨𝐶, 𝐷⟩ ∈ 𝐹 → (𝐶𝐴𝐷𝐵)))
54imp 123 1 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶𝐴𝐷𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2141  cop 3586   × cxp 4609  wf 5194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202
This theorem is referenced by:  feu  5380  fcnvres  5381  fsn  5668
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