![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6382 | . 2 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
4 | 2, 1 | fnsn 6382 | . . 3 ⊢ {〈𝐵, 𝐴〉} Fn {𝐵} |
5 | 1, 2 | cnvsn 6050 | . . . 4 ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
6 | 5 | fneq1i 6420 | . . 3 ⊢ (◡{〈𝐴, 𝐵〉} Fn {𝐵} ↔ {〈𝐵, 𝐴〉} Fn {𝐵}) |
7 | 4, 6 | mpbir 234 | . 2 ⊢ ◡{〈𝐴, 𝐵〉} Fn {𝐵} |
8 | dff1o4 6598 | . 2 ⊢ ({〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} ↔ ({〈𝐴, 𝐵〉} Fn {𝐴} ∧ ◡{〈𝐴, 𝐵〉} Fn {𝐵})) | |
9 | 3, 7, 8 | mpbir2an 710 | 1 ⊢ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 Vcvv 3441 {csn 4525 〈cop 4531 ◡ccnv 5518 Fn wfn 6319 –1-1-onto→wf1o 6323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 |
This theorem is referenced by: f1osng 6630 fsn 6874 ensn1 8556 phplem2 8681 isinf 8715 pssnn 8720 ac6sfi 8746 marypha1lem 8881 hashf1lem1 13809 0ram 16346 mdet0f1o 21198 imasdsf1olem 22980 istrkg2ld 26254 axlowdimlem10 26745 subfacp1lem5 32544 poimirlem3 35060 grposnOLD 35320 |
Copyright terms: Public domain | W3C validator |