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Mirrors > Home > ILE Home > Th. List > f1osn | GIF version |
Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
f1osn.1 | ⊢ 𝐴 ∈ V |
f1osn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
f1osn | ⊢ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1osn.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | f1osn.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | fnsn 4953 | . 2 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
4 | 2, 1 | fnsn 4953 | . . 3 ⊢ {〈𝐵, 𝐴〉} Fn {𝐵} |
5 | 1, 2 | cnvsn 4803 | . . . 4 ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
6 | 5 | fneq1i 4993 | . . 3 ⊢ (◡{〈𝐴, 𝐵〉} Fn {𝐵} ↔ {〈𝐵, 𝐴〉} Fn {𝐵}) |
7 | 4, 6 | mpbir 134 | . 2 ⊢ ◡{〈𝐴, 𝐵〉} Fn {𝐵} |
8 | dff1o4 5134 | . 2 ⊢ ({〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} ↔ ({〈𝐴, 𝐵〉} Fn {𝐴} ∧ ◡{〈𝐴, 𝐵〉} Fn {𝐵})) | |
9 | 3, 7, 8 | mpbir2an 849 | 1 ⊢ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1393 Vcvv 2557 {csn 3375 〈cop 3378 ◡ccnv 4344 Fn wfn 4897 –1-1-onto→wf1o 4901 |
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-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: f1osng 5167 fsn 5335 ensn1 6276 phplem2 6316 ac6sfi 6352 |
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