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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 6152 | . . . . . 6 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | |
2 | 1 | simplbi 493 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) |
3 | brrelex1 5405 | . . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝑦) → 𝐴 ∈ V) | |
4 | 3 | ex 403 | . . . . 5 ⊢ (Rel 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V)) |
6 | 5 | ancrd 547 | . . 3 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦))) |
7 | 6 | alrimiv 1970 | . 2 ⊢ (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦))) |
8 | breq1 4891 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
9 | 8 | mobidv 2564 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦)) |
10 | 9 | imbi2d 332 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦))) |
11 | 1 | simprbi 492 | . . . . . 6 ⊢ (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦) |
12 | 11 | 19.21bi 2173 | . . . . 5 ⊢ (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) |
13 | 10, 12 | vtoclg 3467 | . . . 4 ⊢ (𝐴 ∈ V → (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)) |
14 | 13 | com12 32 | . . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦)) |
15 | moanimv 2654 | . . 3 ⊢ (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦)) | |
16 | 14, 15 | sylibr 226 | . 2 ⊢ (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦)) |
17 | moim 2556 | . 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 386 ∀wal 1599 = wceq 1601 ∈ wcel 2107 ∃*wmo 2549 Vcvv 3398 class class class wbr 4888 Rel wrel 5362 Fun wfun 6131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-fun 6139 |
This theorem is referenced by: funeu 6162 funco 6177 fununmo 6183 imadif 6220 fneu 6243 dff3 6638 shftfn 14226 |
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