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Mirrors > Home > MPE Home > Th. List > fnbr | Structured version Visualization version GIF version |
Description: The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.) |
Ref | Expression |
---|---|
fnbr | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 6650 | . . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | releldm 5940 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹) | |
3 | 1, 2 | sylan 579 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹) |
4 | fndm 6651 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
5 | 4 | eleq2d 2815 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴)) |
6 | 5 | biimpa 476 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ dom 𝐹) → 𝐵 ∈ 𝐴) |
7 | 3, 6 | syldan 590 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 class class class wbr 5142 dom cdm 5672 Rel wrel 5677 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: fnop 6657 dffn5 6951 feqmptdf 6963 dffo4 7107 dffo5 7108 tfrlem5 8394 occllem 31106 chscllem2 31441 tfsconcat0i 42768 brcoffn 43454 fvelima2 44630 dfafn5a 46534 |
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