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Mirrors > Home > ILE Home > Th. List > f1osn | Unicode 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 | |
f1osn.2 |
Ref | Expression |
---|---|
f1osn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1osn.1 | . . 3 | |
2 | f1osn.2 | . . 3 | |
3 | 1, 2 | fnsn 5224 | . 2 |
4 | 2, 1 | fnsn 5224 | . . 3 |
5 | 1, 2 | cnvsn 5068 | . . . 4 |
6 | 5 | fneq1i 5264 | . . 3 |
7 | 4, 6 | mpbir 145 | . 2 |
8 | dff1o4 5422 | . 2 | |
9 | 3, 7, 8 | mpbir2an 927 | 1 |
Colors of variables: wff set class |
Syntax hints: wcel 2128 cvv 2712 csn 3560 cop 3563 ccnv 4585 wfn 5165 wf1o 5169 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 |
This theorem is referenced by: f1osng 5455 fsn 5639 mapsn 6635 ensn1 6741 phplem2 6798 ac6sfi 6843 fxnn0nninf 10337 |
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