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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 6536 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | |
2 | 1 | sseld 3968 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝐷〉 ∈ 𝐹 → 〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵))) |
3 | opelxp 5593 | . . 3 ⊢ (〈𝐶, 𝐷〉 ∈ (𝐴 × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
4 | 2, 3 | syl6ib 253 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (〈𝐶, 𝐷〉 ∈ 𝐹 → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))) |
5 | 4 | imp 409 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 〈𝐶, 𝐷〉 ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 〈cop 4575 × cxp 5555 ⟶wf 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 |
This theorem is referenced by: feu 6556 fcnvres 6558 fsn 6899 |
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