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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 6573 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | |
2 | 1 | sseld 3900 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝐷〉 ∈ 𝐹 → 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵))) |
3 | opelxp 5587 | . . 3 ⊢ (〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
4 | 2, 3 | syl6ib 254 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝐷〉 ∈ 𝐹 → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))) |
5 | 4 | imp 410 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝐷〉 ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2110 〈cop 4547 × cxp 5549 ⟶wf 6376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-dm 5561 df-rn 5562 df-fun 6382 df-fn 6383 df-f 6384 |
This theorem is referenced by: feu 6595 fcnvres 6596 fsn 6950 |
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