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Theorem opelf 5341
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 5337 . . . 4 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
21sseld 3127 . . 3 (𝐹:𝐴𝐵 → (⟨𝐶, 𝐷⟩ ∈ 𝐹 → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)))
3 opelxp 4616 . . 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 2128  cop 3563   × cxp 4584  wf 5166
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-cnv 4594  df-dm 4596  df-rn 4597  df-fun 5172  df-fn 5173  df-f 5174
This theorem is referenced by:  feu  5352  fcnvres  5353  fsn  5639
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