![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > dffun5r | GIF version |
Description: A way of proving a relation is a function, analogous to mo2r 1949. (Contributed by Jim Kingdon, 27-May-2020.) |
Ref | Expression |
---|---|
dffun5r | ⊢ ((Rel A ∧ ∀x∃z∀y(〈x, y〉 ∈ A → y = z)) → Fun A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1418 | . . . . . 6 ⊢ Ⅎz〈x, y〉 ∈ A | |
2 | 1 | mo2r 1949 | . . . . 5 ⊢ (∃z∀y(〈x, y〉 ∈ A → y = z) → ∃*y〈x, y〉 ∈ A) |
3 | opeq2 3541 | . . . . . . 7 ⊢ (y = z → 〈x, y〉 = 〈x, z〉) | |
4 | 3 | eleq1d 2103 | . . . . . 6 ⊢ (y = z → (〈x, y〉 ∈ A ↔ 〈x, z〉 ∈ A)) |
5 | 4 | mo4 1958 | . . . . 5 ⊢ (∃*y〈x, y〉 ∈ A ↔ ∀y∀z((〈x, y〉 ∈ A ∧ 〈x, z〉 ∈ A) → y = z)) |
6 | 2, 5 | sylib 127 | . . . 4 ⊢ (∃z∀y(〈x, y〉 ∈ A → y = z) → ∀y∀z((〈x, y〉 ∈ A ∧ 〈x, z〉 ∈ A) → y = z)) |
7 | 6 | alimi 1341 | . . 3 ⊢ (∀x∃z∀y(〈x, y〉 ∈ A → y = z) → ∀x∀y∀z((〈x, y〉 ∈ A ∧ 〈x, z〉 ∈ A) → y = z)) |
8 | 7 | anim2i 324 | . 2 ⊢ ((Rel A ∧ ∀x∃z∀y(〈x, y〉 ∈ A → y = z)) → (Rel A ∧ ∀x∀y∀z((〈x, y〉 ∈ A ∧ 〈x, z〉 ∈ A) → y = z))) |
9 | dffun4 4856 | . 2 ⊢ (Fun A ↔ (Rel A ∧ ∀x∀y∀z((〈x, y〉 ∈ A ∧ 〈x, z〉 ∈ A) → y = z))) | |
10 | 8, 9 | sylibr 137 | 1 ⊢ ((Rel A ∧ ∀x∃z∀y(〈x, y〉 ∈ A → y = z)) → Fun A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1240 ∃wex 1378 ∈ wcel 1390 ∃*wmo 1898 〈cop 3370 Rel wrel 4293 Fun wfun 4839 |
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-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-cnv 4296 df-co 4297 df-fun 4847 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |