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Mirrors > Home > MPE Home > Th. List > funeu2 | Structured version Visualization version 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 4584 | . 2 ⊢ (𝐴𝐹𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹) | |
2 | funeu 5828 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦) | |
3 | df-br 4584 | . . . 4 ⊢ (𝐴𝐹𝑦 ↔ 〈𝐴, 𝑦〉 ∈ 𝐹) | |
4 | 3 | eubii 2480 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
5 | 2, 4 | sylib 207 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
6 | 1, 5 | sylan2br 492 | 1 ⊢ ((Fun 𝐹 ∧ 〈𝐴, 𝐵〉 ∈ 𝐹) → ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∃!weu 2458 〈cop 4131 class class class wbr 4583 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-fun 5806 |
This theorem is referenced by: funssres 5844 |
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