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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 6624 | . 2 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
| 4 | 2, 1 | fnsn 6624 | . . 3 ⊢ {〈𝐵, 𝐴〉} Fn {𝐵} |
| 5 | 1, 2 | cnvsn 6246 | . . . 4 ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
| 6 | 5 | fneq1i 6665 | . . 3 ⊢ (◡{〈𝐴, 𝐵〉} Fn {𝐵} ↔ {〈𝐵, 𝐴〉} Fn {𝐵}) |
| 7 | 4, 6 | mpbir 231 | . 2 ⊢ ◡{〈𝐴, 𝐵〉} Fn {𝐵} |
| 8 | dff1o4 6856 | . 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 2108 Vcvv 3480 {csn 4626 〈cop 4632 ◡ccnv 5684 Fn wfn 6556 –1-1-onto→wf1o 6560 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 |
| This theorem is referenced by: f1osng 6889 fsn 7155 ensn1 9061 pssnn 9208 phplem2OLD 9255 isinf 9296 isinfOLD 9297 ac6sfi 9320 marypha1lem 9473 hashf1lem1 14494 0ram 17058 mdet0f1o 22599 imasdsf1olem 24383 istrkg2ld 28468 axlowdimlem10 28966 subfacp1lem5 35189 poimirlem3 37630 grposnOLD 37889 |
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