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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 6609 | . . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | releldm 5904 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹) | |
3 | 1, 2 | sylan 581 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹) |
4 | fndm 6610 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
5 | 4 | eleq2d 2824 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴)) |
6 | 5 | biimpa 478 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ dom 𝐹) → 𝐵 ∈ 𝐴) |
7 | 3, 6 | syldan 592 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 class class class wbr 5110 dom cdm 5638 Rel wrel 5643 Fn wfn 6496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 df-dm 5648 df-fun 6503 df-fn 6504 |
This theorem is referenced by: fnop 6616 dffn5 6906 feqmptdf 6917 dffo4 7058 dffo5 7059 tfrlem5 8331 occllem 30287 chscllem2 30622 brcoffn 42376 fvelima2 43562 dfafn5a 45466 |
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