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Mirrors > Home > ILE Home > Th. List > opelf | 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 5260 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | |
2 | 1 | sseld 3066 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝐷〉 ∈ 𝐹 → 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵))) |
3 | opelxp 4539 | . . 3 ⊢ (〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
4 | 2, 3 | syl6ib 160 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝐷〉 ∈ 𝐹 → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))) |
5 | 4 | imp 123 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝐷〉 ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1465 〈cop 3500 × cxp 4507 ⟶wf 5089 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 |
This theorem is referenced by: feu 5275 fcnvres 5276 fsn 5560 |
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