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Mirrors > Home > MPE Home > Th. List > fnsn | Structured version Visualization version GIF version |
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
fnsn.1 | ⊢ 𝐴 ∈ V |
fnsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
fnsn | ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | fnsn.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | fnsng 5852 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {〈𝐴, 𝐵〉} Fn {𝐴}) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 Fn wfn 5799 |
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 df-fn 5807 |
This theorem is referenced by: f1osn 6088 fnsnb 6337 fvsnun2 6354 elixpsn 7833 axdc3lem4 9158 hashf1lem1 13096 axlowdimlem8 25629 axlowdimlem9 25630 axlowdimlem11 25632 axlowdimlem12 25633 eupath2lem3 26506 bnj927 30093 cvmliftlem4 30524 cvmliftlem5 30525 finixpnum 32564 poimirlem3 32582 |
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