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Mirrors > Home > MPE Home > Th. List > f1osn | Structured version Visualization version 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 6626 | . 2 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
4 | 2, 1 | fnsn 6626 | . . 3 ⊢ {〈𝐵, 𝐴〉} Fn {𝐵} |
5 | 1, 2 | cnvsn 6248 | . . . 4 ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
6 | 5 | fneq1i 6666 | . . 3 ⊢ (◡{〈𝐴, 𝐵〉} Fn {𝐵} ↔ {〈𝐵, 𝐴〉} Fn {𝐵}) |
7 | 4, 6 | mpbir 231 | . 2 ⊢ ◡{〈𝐴, 𝐵〉} Fn {𝐵} |
8 | dff1o4 6857 | . 2 ⊢ ({〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} ↔ ({〈𝐴, 𝐵〉} Fn {𝐴} ∧ ◡{〈𝐴, 𝐵〉} Fn {𝐵})) | |
9 | 3, 7, 8 | mpbir2an 711 | 1 ⊢ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 ◡ccnv 5688 Fn wfn 6558 –1-1-onto→wf1o 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: f1osng 6890 fsn 7155 ensn1 9060 pssnn 9207 phplem2OLD 9253 isinf 9294 isinfOLD 9295 ac6sfi 9318 marypha1lem 9471 hashf1lem1 14491 0ram 17054 mdet0f1o 22615 imasdsf1olem 24399 istrkg2ld 28483 axlowdimlem10 28981 subfacp1lem5 35169 poimirlem3 37610 grposnOLD 37869 |
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