Theorem opelf 5457

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:𝐴⟶𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))

Proof of Theorem opelf

Step	Hyp	Ref	Expression
1		fssxp 5452	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
2	1	sseld 3196	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝐶, 𝐷⟩ ∈ 𝐹 → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)))
3		opelxp 4712	. . 3 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
4	2, 3	imbitrdi 161	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝐶, 𝐷⟩ ∈ 𝐹 → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)))
5	4	imp 124	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2177  ⟨cop 3640   × cxp 4680  ⟶wf 5275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-pr 4260

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-br 4051  df-opab 4113  df-xp 4688  df-rel 4689  df-cnv 4690  df-dm 4692  df-rn 4693  df-fun 5281  df-fn 5282  df-f 5283

This theorem is referenced by:  feu  5469  fcnvres  5470  fsn  5764