Theorem opelf 5005

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 5001	. . . 4 ⊢ (𝐹:A⟶B → 𝐹 ⊆ (A × B))
2	1	sseld 2938	. . 3 ⊢ (𝐹:A⟶B → (⟨𝐶, 𝐷⟩ ∈ 𝐹 → ⟨𝐶, 𝐷⟩ ∈ (A × B)))
3		opelxp 4317	. . 3 ⊢ (⟨𝐶, 𝐷⟩ ∈ (A × B) ↔ (𝐶 ∈ A ∧ 𝐷 ∈ B))
4	2, 3	syl6ib 150	. 2 ⊢ (𝐹:A⟶B → (⟨𝐶, 𝐷⟩ ∈ 𝐹 → (𝐶 ∈ A ∧ 𝐷 ∈ B)))
5	4	imp 115	1 ⊢ ((𝐹:A⟶B ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶 ∈ A ∧ 𝐷 ∈ B))

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  ⟨cop 3370   × cxp 4286  ⟶wf 4841

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849

This theorem is referenced by:  feu  5015  fcnvres  5016  fsn  5278