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Mirrors > Home > ILE Home > Th. List > fnsng | GIF version |
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
fnsng | ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → {〈A, B〉} Fn {A}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funsng 4889 | . 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → Fun {〈A, B〉}) | |
2 | dmsnopg 4735 | . . 3 ⊢ (B ∈ 𝑊 → dom {〈A, B〉} = {A}) | |
3 | 2 | adantl 262 | . 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → dom {〈A, B〉} = {A}) |
4 | df-fn 4848 | . 2 ⊢ ({〈A, B〉} Fn {A} ↔ (Fun {〈A, B〉} ∧ dom {〈A, B〉} = {A})) | |
5 | 1, 3, 4 | sylanbrc 394 | 1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → {〈A, B〉} Fn {A}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 {csn 3367 〈cop 3370 dom cdm 4288 Fun wfun 4839 Fn wfn 4840 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-fun 4847 df-fn 4848 |
This theorem is referenced by: fnsn 4896 fnunsn 4949 fsnunfv 5306 tfr0 5878 |
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