![]() |
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 2052. (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 1509 | . . . . . 6 ⊢ Ⅎ𝑧〈𝑥, 𝑦〉 ∈ 𝐴 | |
2 | 1 | mo2r 2052 | . . . . 5 ⊢ (∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧) → ∃*𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
3 | opeq2 3714 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑧〉) | |
4 | 3 | eleq1d 2209 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑧〉 ∈ 𝐴)) |
5 | 4 | mo4 2061 | . . . . 5 ⊢ (∃*𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧)) |
6 | 2, 5 | sylib 121 | . . . 4 ⊢ (∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧) → ∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧)) |
7 | 6 | alimi 1432 | . . 3 ⊢ (∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧) → ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧)) |
8 | 7 | anim2i 340 | . 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 5142 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧))) | |
10 | 8, 9 | sylibr 133 | 1 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧)) → Fun 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1330 ∃wex 1469 ∈ wcel 1481 ∃*wmo 2001 〈cop 3535 Rel wrel 4552 Fun wfun 5125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-id 4223 df-cnv 4555 df-co 4556 df-fun 5133 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |