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Mirrors > Home > ILE Home > Th. List > f1osng | 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 3538 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | f1oeq2 5357 | . . . 4 ⊢ ({𝑎} = {𝐴} → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
4 | opeq1 3705 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
5 | 4 | sneqd 3540 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
6 | f1oeq1 5356 | . . . 4 ⊢ ({〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉} → ({〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
8 | 3, 7 | bitrd 187 | . 2 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
9 | sneq 3538 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) | |
10 | f1oeq3 5358 | . . . 4 ⊢ ({𝑏} = {𝐵} → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵})) | |
11 | 9, 10 | syl 14 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵})) |
12 | opeq2 3706 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
13 | 12 | sneqd 3540 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
14 | f1oeq1 5356 | . . . 4 ⊢ ({〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉} → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) | |
15 | 13, 14 | syl 14 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) |
16 | 11, 15 | bitrd 187 | . 2 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) |
17 | vex 2689 | . . 3 ⊢ 𝑎 ∈ V | |
18 | vex 2689 | . . 3 ⊢ 𝑏 ∈ V | |
19 | 17, 18 | f1osn 5407 | . 2 ⊢ {〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} |
20 | 8, 16, 19 | vtocl2g 2750 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {csn 3527 〈cop 3530 –1-1-onto→wf1o 5122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 |
This theorem is referenced by: f1sng 5409 f1oprg 5411 fsnunf 5620 dif1en 6773 1fv 9916 zfz1isolem1 10583 sumsnf 11178 ennnfonelemhf1o 11926 |
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