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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 4636 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
| 2 | 1 | f1oeq2d 6844 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
| 3 | opeq1 4873 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
| 4 | 3 | sneqd 4638 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
| 5 | 4 | f1oeq1d 6843 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
| 6 | 2, 5 | bitrd 279 | . 2 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
| 7 | sneq 4636 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) | |
| 8 | 7 | f1oeq3d 6845 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵})) |
| 9 | opeq2 4874 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
| 10 | 9 | sneqd 4638 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
| 11 | 10 | f1oeq1d 6843 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) |
| 12 | 8, 11 | bitrd 279 | . 2 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) |
| 13 | vex 3484 | . . 3 ⊢ 𝑎 ∈ V | |
| 14 | vex 3484 | . . 3 ⊢ 𝑏 ∈ V | |
| 15 | 13, 14 | f1osn 6888 | . 2 ⊢ {〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} |
| 16 | 6, 12, 15 | vtocl2g 3574 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 –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: f1sng 6890 f1oprswap 6892 f1oprg 6893 f1o2sn 7162 fsnunf 7205 fsnex 7303 suppsnop 8203 mapsnd 8926 ralxpmap 8936 en2sn 9081 enfixsn 9121 fseqenlem1 10064 canthp1lem2 10693 sumsnf 15779 prodsn 15998 prodsnf 16000 vdwlem8 17026 gsumws1 18851 symg1bas 19408 dprdsn 20056 eupthp1 30235 s1f1 32927 poimirlem16 37643 poimirlem17 37644 poimirlem19 37646 poimirlem20 37647 metakunt25 42230 mapfzcons 42727 sumsnd 45031 1hegrlfgr 48048 |
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