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Mirrors > Home > ILE Home > Th. List > dffun5r | GIF version |
Description: A way of proving a relation is a function, analogous to mo2r 1995. (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 1462 | . . . . . 6 ⊢ Ⅎ𝑧〈𝑥, 𝑦〉 ∈ 𝐴 | |
2 | 1 | mo2r 1995 | . . . . 5 ⊢ (∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧) → ∃*𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
3 | opeq2 3597 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑧〉) | |
4 | 3 | eleq1d 2151 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑧〉 ∈ 𝐴)) |
5 | 4 | mo4 2004 | . . . . 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 1385 | . . 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 4979 | . 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 1283 ∃wex 1422 ∈ wcel 1434 ∃*wmo 1944 〈cop 3425 Rel wrel 4405 Fun wfun 4962 |
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 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-br 3812 df-opab 3866 df-id 4083 df-cnv 4408 df-co 4409 df-fun 4970 |
This theorem is referenced by: (None) |
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