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Mirrors > Home > ILE Home > Th. List > funfveu | GIF version |
Description: A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Ref | Expression |
---|---|
funfveu | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2200 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹)) | |
2 | 1 | anbi2d 459 | . . . 4 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))) |
3 | breq1 3927 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
4 | 3 | eubidv 2005 | . . . 4 ⊢ (𝑥 = 𝐴 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦)) |
5 | 2, 4 | imbi12d 233 | . . 3 ⊢ (𝑥 = 𝐴 → (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ∃!𝑦 𝑥𝐹𝑦) ↔ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦))) |
6 | dffun8 5146 | . . . . 5 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∃!𝑦 𝑥𝐹𝑦)) | |
7 | 6 | simprbi 273 | . . . 4 ⊢ (Fun 𝐹 → ∀𝑥 ∈ dom 𝐹∃!𝑦 𝑥𝐹𝑦) |
8 | 7 | r19.21bi 2518 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ∃!𝑦 𝑥𝐹𝑦) |
9 | 5, 8 | vtoclg 2741 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)) |
10 | 9 | anabsi7 570 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∃!weu 1997 ∀wral 2414 class class class wbr 3924 dom cdm 4534 Rel wrel 4539 Fun wfun 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-cnv 4542 df-co 4543 df-dm 4544 df-fun 5120 |
This theorem is referenced by: funfvex 5431 |
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