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| Description: The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.) | 
| Ref | Expression | 
|---|---|
| fnop | ⊢ ((𝐹 Fn 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝐹) → 𝐵 ∈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-br 5143 | . 2 ⊢ (𝐵𝐹𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐹) | |
| 2 | fnbr 6675 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴) | |
| 3 | 1, 2 | sylan2br 595 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝐹) → 𝐵 ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 〈cop 4631 class class class wbr 5142 Fn wfn 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-dm 5694 df-fun 6562 df-fn 6563 | 
| This theorem is referenced by: 2elresin 6688 wfrlem12OLD 8361 tfrlem9 8426 wlkp1lem2 29693 poimirlem4 37632 | 
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