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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 6670 | . . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
| 2 | releldm 5955 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹) | |
| 3 | 1, 2 | sylan 580 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹) |
| 4 | fndm 6671 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
| 5 | 4 | eleq2d 2827 | . . 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 591 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 dom cdm 5685 Rel wrel 5690 Fn wfn 6556 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: fnop 6677 fvelima2 6961 dffn5 6967 feqmptdf 6979 dffo4 7123 dffo5 7124 tfrlem5 8420 occllem 31322 chscllem2 31657 tfsconcat0i 43358 brcoffn 44043 dfafn5a 47172 |
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