Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > opelf | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
opelf | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝐷〉 ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fssxp 5973 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | |
2 | 1 | sseld 3567 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝐷〉 ∈ 𝐹 → 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵))) |
3 | opelxp 5070 | . . 3 ⊢ (〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
4 | 2, 3 | syl6ib 240 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝐷〉 ∈ 𝐹 → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))) |
5 | 4 | imp 444 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝐷〉 ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 〈cop 4131 × cxp 5036 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 |
This theorem is referenced by: feu 5993 fcnvres 5995 fsn 6308 |
Copyright terms: Public domain | W3C validator |