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| Mirrors > Home > ILE Home > Th. List > fnbr | 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 5372 | . . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
| 2 | releldm 4913 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹) | |
| 3 | 1, 2 | sylan 283 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹) |
| 4 | fndm 5373 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
| 5 | 4 | eleq2d 2275 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴)) |
| 6 | 5 | biimpa 296 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ dom 𝐹) → 𝐵 ∈ 𝐴) |
| 7 | 3, 6 | syldan 282 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2176 class class class wbr 4044 dom cdm 4675 Rel wrel 4680 Fn wfn 5266 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-br 4045 df-opab 4106 df-xp 4681 df-rel 4682 df-dm 4685 df-fun 5273 df-fn 5274 |
| This theorem is referenced by: fnop 5379 dffn5im 5624 dffo4 5728 dffo5 5729 tfrlem5 6400 |
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