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Mirrors > Home > ILE Home > Th. List > funsn | GIF version |
Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.) |
Ref | Expression |
---|---|
funsn.1 | ⊢ 𝐴 ∈ V |
funsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
funsn | ⊢ Fun {〈𝐴, 𝐵〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | funsn.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | funsng 5094 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {〈𝐴, 𝐵〉}) | |
4 | 1, 2, 3 | mp2an 418 | 1 ⊢ Fun {〈𝐴, 𝐵〉} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1445 Vcvv 2633 {csn 3466 〈cop 3469 Fun wfun 5043 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-fun 5051 |
This theorem is referenced by: funtp 5101 fun0 5106 fvsn 5531 |
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