![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 5333 | . . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | releldm 4880 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹) | |
3 | 1, 2 | sylan 283 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹) |
4 | fndm 5334 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
5 | 4 | eleq2d 2259 | . . 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 2160 class class class wbr 4018 dom cdm 4644 Rel wrel 4649 Fn wfn 5230 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-dm 4654 df-fun 5237 df-fn 5238 |
This theorem is referenced by: fnop 5338 dffn5im 5581 dffo4 5684 dffo5 5685 tfrlem5 6338 |
Copyright terms: Public domain | W3C validator |