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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 4409 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | f1oeq2d 6378 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
3 | opeq1 4625 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
4 | 3 | sneqd 4411 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
5 | f1oeq1 6371 | . . . 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 271 | . 2 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
8 | sneq 4409 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) | |
9 | f1oeq3 6373 | . . . 4 ⊢ ({𝑏} = {𝐵} → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵})) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵})) |
11 | opeq2 4626 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
12 | 11 | sneqd 4411 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
13 | f1oeq1 6371 | . . . 4 ⊢ ({〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉} → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) |
15 | 10, 14 | bitrd 271 | . 2 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) |
16 | vex 3417 | . . 3 ⊢ 𝑎 ∈ V | |
17 | vex 3417 | . . 3 ⊢ 𝑏 ∈ V | |
18 | 16, 17 | f1osn 6421 | . 2 ⊢ {〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} |
19 | 7, 15, 18 | vtocl2g 3486 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 {csn 4399 〈cop 4405 –1-1-onto→wf1o 6126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 |
This theorem is referenced by: f1sng 6423 f1oprswap 6425 f1oprg 6426 f1o2sn 6663 fsnunf 6712 fsnex 6798 suppsnop 7578 mapsnd 8170 ralxpmap 8180 enfixsn 8344 fseqenlem1 9167 canthp1lem2 9797 sumsnf 14857 prodsn 15072 prodsnf 15074 vdwlem8 16070 gsumws1 17736 symg1bas 18173 dprdsn 18796 eupthp1 27589 poimirlem16 33968 poimirlem17 33969 poimirlem19 33971 poimirlem20 33972 mapfzcons 38122 sumsnd 40002 1hegrlfgr 42578 |
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