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Mirrors > Home > ILE Home > Th. List > funeu2 | GIF version |
Description: There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.) |
Ref | Expression |
---|---|
funeu2 | ⊢ ((Fun 𝐹 ∧ 〈𝐴, 𝐵〉 ∈ 𝐹) → ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4004 | . 2 ⊢ (𝐴𝐹𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹) | |
2 | funeu 5241 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦) | |
3 | df-br 4004 | . . . 4 ⊢ (𝐴𝐹𝑦 ↔ 〈𝐴, 𝑦〉 ∈ 𝐹) | |
4 | 3 | eubii 2035 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
5 | 2, 4 | sylib 122 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
6 | 1, 5 | sylan2br 288 | 1 ⊢ ((Fun 𝐹 ∧ 〈𝐴, 𝐵〉 ∈ 𝐹) → ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∃!weu 2026 ∈ wcel 2148 〈cop 3595 class class class wbr 4003 Fun wfun 5210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-fun 5218 |
This theorem is referenced by: funssres 5258 |
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