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Mirrors > Home > ILE Home > Th. List > funmo | GIF version |
Description: A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) |
Ref | Expression |
---|---|
funmo | ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun6 4916 | . . . . . 6 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | |
2 | 1 | simplbi 259 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) |
3 | brrelex 4382 | . . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝑦) → 𝐴 ∈ V) | |
4 | 3 | ex 108 | . . . . 5 ⊢ (Rel 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V)) |
5 | 2, 4 | syl 14 | . . . 4 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V)) |
6 | 5 | ancrd 309 | . . 3 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦))) |
7 | 6 | alrimiv 1754 | . 2 ⊢ (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦))) |
8 | breq1 3767 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
9 | 8 | mobidv 1936 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦)) |
10 | 9 | imbi2d 219 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦))) |
11 | 1 | simprbi 260 | . . . . . 6 ⊢ (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦) |
12 | 11 | 19.21bi 1450 | . . . . 5 ⊢ (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) |
13 | 10, 12 | vtoclg 2613 | . . . 4 ⊢ (𝐴 ∈ V → (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)) |
14 | 13 | com12 27 | . . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦)) |
15 | moanimv 1975 | . . 3 ⊢ (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦)) | |
16 | 14, 15 | sylibr 137 | . 2 ⊢ (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦)) |
17 | moim 1964 | . 2 ⊢ (∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)) → (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) → ∃*𝑦 𝐴𝐹𝑦)) | |
18 | 7, 16, 17 | sylc 56 | 1 ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 = wceq 1243 ∈ wcel 1393 ∃*wmo 1901 Vcvv 2557 class class class wbr 3764 Rel wrel 4350 Fun wfun 4896 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-fun 4904 |
This theorem is referenced by: funeu 4926 funco 4940 imadif 4979 fneu 5003 dff3im 5312 shftfn 9425 |
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