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Mirrors > Home > ILE Home > Th. List > fnsn | 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 5140 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {〈𝐴, 𝐵〉} Fn {𝐴}) | |
4 | 1, 2, 3 | mp2an 422 | 1 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 Vcvv 2660 {csn 3497 〈cop 3500 Fn wfn 5088 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-fun 5095 df-fn 5096 |
This theorem is referenced by: f1osn 5375 fvsnun2 5586 elixpsn 6597 |
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