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Mirrors > Home > MPE Home > Th. List > fnop | Structured version Visualization version GIF version |
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 6656 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴) | |
3 | 1, 2 | sylan2br 594 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝐹) → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 〈cop 4630 class class class wbr 5142 Fn wfn 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-dm 5682 df-fun 6544 df-fn 6545 |
This theorem is referenced by: 2elresin 6670 wfrlem12OLD 8334 tfrlem9 8399 wlkp1lem2 29481 poimirlem4 37091 |
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