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Mirrors > Home > MPE Home > Th. List > funmo | Structured version Visualization version 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 5903 | . . . . . 6 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | |
2 | 1 | simplbi 476 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) |
3 | brrelex 5156 | . . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝑦) → 𝐴 ∈ V) | |
4 | 3 | ex 450 | . . . . 5 ⊢ (Rel 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V)) |
6 | 5 | ancrd 577 | . . 3 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦))) |
7 | 6 | alrimiv 1855 | . 2 ⊢ (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦))) |
8 | breq1 4656 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
9 | 8 | mobidv 2491 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦)) |
10 | 9 | imbi2d 330 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦))) |
11 | 1 | simprbi 480 | . . . . . 6 ⊢ (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦) |
12 | 11 | 19.21bi 2059 | . . . . 5 ⊢ (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) |
13 | 10, 12 | vtoclg 3266 | . . . 4 ⊢ (𝐴 ∈ V → (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)) |
14 | 13 | com12 32 | . . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦)) |
15 | moanimv 2531 | . . 3 ⊢ (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦)) | |
16 | 14, 15 | sylibr 224 | . 2 ⊢ (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦)) |
17 | moim 2519 | . 2 ⊢ (∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)) → (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) → ∃*𝑦 𝐴𝐹𝑦)) | |
18 | 7, 16, 17 | sylc 65 | 1 ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ∃*wmo 2471 Vcvv 3200 class class class wbr 4653 Rel wrel 5119 Fun wfun 5882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-fun 5890 |
This theorem is referenced by: funeu 5913 funco 5928 fununmo 5933 imadif 5973 fneu 5995 dff3 6372 shftfn 13813 |
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