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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 4973 | . 2 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
4 | 2, 1 | fnsn 4973 | . . 3 ⊢ {〈𝐵, 𝐴〉} Fn {𝐵} |
5 | 1, 2 | cnvsn 4823 | . . . 4 ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
6 | 5 | fneq1i 5013 | . . 3 ⊢ (◡{〈𝐴, 𝐵〉} Fn {𝐵} ↔ {〈𝐵, 𝐴〉} Fn {𝐵}) |
7 | 4, 6 | mpbir 144 | . 2 ⊢ ◡{〈𝐴, 𝐵〉} Fn {𝐵} |
8 | dff1o4 5154 | . 2 ⊢ ({〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} ↔ ({〈𝐴, 𝐵〉} Fn {𝐴} ∧ ◡{〈𝐴, 𝐵〉} Fn {𝐵})) | |
9 | 3, 7, 8 | mpbir2an 883 | 1 ⊢ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1433 Vcvv 2601 {csn 3398 〈cop 3401 ◡ccnv 4362 Fn wfn 4917 –1-1-onto→wf1o 4921 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 |
This theorem is referenced by: f1osng 5187 fsn 5356 ensn1 6299 phplem2 6339 ac6sfi 6379 |
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