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Mirrors > Home > MPE Home > Th. List > f1osng | Structured version Visualization version GIF version |
Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.) |
Ref | Expression |
---|---|
f1osng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4535 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | f1oeq2d 6586 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
3 | opeq1 4763 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
4 | 3 | sneqd 4537 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
5 | f1oeq1 6579 | . . . 4 ⊢ ({〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉} → ({〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
7 | 2, 6 | bitrd 282 | . 2 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
8 | sneq 4535 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) | |
9 | 8 | f1oeq3d 6587 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵})) |
10 | opeq2 4765 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
11 | 10 | sneqd 4537 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
12 | f1oeq1 6579 | . . . 4 ⊢ ({〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉} → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) |
14 | 9, 13 | bitrd 282 | . 2 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) |
15 | vex 3444 | . . 3 ⊢ 𝑎 ∈ V | |
16 | vex 3444 | . . 3 ⊢ 𝑏 ∈ V | |
17 | 15, 16 | f1osn 6629 | . 2 ⊢ {〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} |
18 | 7, 14, 17 | vtocl2g 3520 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 〈cop 4531 –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: f1sng 6631 f1oprswap 6633 f1oprg 6634 f1o2sn 6881 fsnunf 6924 fsnex 7017 suppsnop 7827 mapsnd 8433 ralxpmap 8443 enfixsn 8609 fseqenlem1 9435 canthp1lem2 10064 sumsnf 15091 prodsn 15308 prodsnf 15310 vdwlem8 16314 gsumws1 17994 symg1bas 18511 dprdsn 19151 eupthp1 28001 s1f1 30645 poimirlem16 35073 poimirlem17 35074 poimirlem19 35076 poimirlem20 35077 metakunt25 39374 mapfzcons 39657 sumsnd 41655 1hegrlfgr 44360 |
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