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Theorem opelf 6227
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 6222 . . . 4 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
21sseld 3744 . . 3 (𝐹:𝐴𝐵 → (⟨𝐶, 𝐷⟩ ∈ 𝐹 → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)))
3 opelxp 5304 . . 3 (⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵) ↔ (𝐶𝐴𝐷𝐵))
42, 3syl6ib 241 . 2 (𝐹:𝐴𝐵 → (⟨𝐶, 𝐷⟩ ∈ 𝐹 → (𝐶𝐴𝐷𝐵)))
54imp 444 1 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2140  cop 4328   × cxp 5265  wf 6046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-rel 5274  df-cnv 5275  df-dm 5277  df-rn 5278  df-fun 6052  df-fn 6053  df-f 6054
This theorem is referenced by:  feu  6242  fcnvres  6244  fsn  6567
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