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Mirrors > Home > ILE Home > Th. List > funsng | GIF version |
Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.) |
Ref | Expression |
---|---|
funsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcnvsn 4945 | . 2 ⊢ Fun ◡{〈𝐵, 𝐴〉} | |
2 | cnvsng 4806 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ◡{〈𝐵, 𝐴〉} = {〈𝐴, 𝐵〉}) | |
3 | 2 | ancoms 255 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐵, 𝐴〉} = {〈𝐴, 𝐵〉}) |
4 | 3 | funeqd 4923 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Fun ◡{〈𝐵, 𝐴〉} ↔ Fun {〈𝐴, 𝐵〉})) |
5 | 1, 4 | mpbii 136 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 {csn 3375 〈cop 3378 ◡ccnv 4344 Fun wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-fun 4904 |
This theorem is referenced by: fnsng 4947 funsn 4948 funprg 4949 funtpg 4950 |
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