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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 3786 | . 2 ⊢ (𝐴𝐹𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹) | |
2 | funeu 4946 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦) | |
3 | df-br 3786 | . . . 4 ⊢ (𝐴𝐹𝑦 ↔ 〈𝐴, 𝑦〉 ∈ 𝐹) | |
4 | 3 | eubii 1950 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
5 | 2, 4 | sylib 120 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
6 | 1, 5 | sylan2br 282 | 1 ⊢ ((Fun 𝐹 ∧ 〈𝐴, 𝐵〉 ∈ 𝐹) → ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1433 ∃!weu 1941 〈cop 3401 class class class wbr 3785 Fun wfun 4916 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-fun 4924 |
This theorem is referenced by: funssres 4962 |
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