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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.) (Proof shortened by SN, 19-Dec-2024.) | 
| Ref | Expression | 
|---|---|
| funmo | ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dffun6 6574 | . . . . . 6 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | |
| 2 | 1 | simplbi 497 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | 
| 3 | brrelex1 5738 | . . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝑦) → 𝐴 ∈ V) | |
| 4 | 3 | ex 412 | . . . . 5 ⊢ (Rel 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V)) | 
| 5 | 2, 4 | syl 17 | . . . 4 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V)) | 
| 6 | 5 | ancrd 551 | . . 3 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦))) | 
| 7 | 6 | alrimiv 1927 | . 2 ⊢ (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦))) | 
| 8 | 1 | simprbi 496 | . . . 4 ⊢ (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦) | 
| 9 | breq1 5146 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
| 10 | 9 | mobidv 2549 | . . . . 5 ⊢ (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦)) | 
| 11 | 10 | spcgv 3596 | . . . 4 ⊢ (𝐴 ∈ V → (∀𝑥∃*𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦)) | 
| 12 | 8, 11 | syl5com 31 | . . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦)) | 
| 13 | moanimv 2619 | . . 3 ⊢ (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦)) | |
| 14 | 12, 13 | sylibr 234 | . 2 ⊢ (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦)) | 
| 15 | moim 2544 | . 2 ⊢ (∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)) → (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) → ∃*𝑦 𝐴𝐹𝑦)) | |
| 16 | 7, 14, 15 | sylc 65 | 1 ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∃*wmo 2538 Vcvv 3480 class class class wbr 5143 Rel wrel 5690 Fun wfun 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 | 
| This theorem is referenced by: funeu 6591 funco 6606 fununmo 6613 imadif 6650 fneu 6678 dff3 7120 shftfn 15112 | 
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