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 1993. (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 1461 | . . . . . 6 ⊢ Ⅎ𝑧〈𝑥, 𝑦〉 ∈ 𝐴 | |
2 | 1 | mo2r 1993 | . . . . 5 ⊢ (∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧) → ∃*𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
3 | opeq2 3571 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑧〉) | |
4 | 3 | eleq1d 2147 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑧〉 ∈ 𝐴)) |
5 | 4 | mo4 2002 | . . . . 5 ⊢ (∃*𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧)) |
6 | 2, 5 | sylib 120 | . . . 4 ⊢ (∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧) → ∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧)) |
7 | 6 | alimi 1384 | . . 3 ⊢ (∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧) → ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧)) |
8 | 7 | anim2i 334 | . 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 4933 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧))) | |
10 | 8, 9 | sylibr 132 | 1 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧)) → Fun 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1282 ∃wex 1421 ∈ wcel 1433 ∃*wmo 1942 〈cop 3401 Rel wrel 4368 Fun wfun 4916 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-cnv 4371 df-co 4372 df-fun 4924 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |