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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 4634 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | f1oeq2d 6829 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
3 | opeq1 4869 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
4 | 3 | sneqd 4636 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
5 | 4 | f1oeq1d 6828 | . . 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 4634 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) | |
8 | 7 | f1oeq3d 6830 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵})) |
9 | opeq2 4870 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
10 | 9 | sneqd 4636 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
11 | 10 | f1oeq1d 6828 | . . 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 3473 | . . 3 ⊢ 𝑎 ∈ V | |
14 | vex 3473 | . . 3 ⊢ 𝑏 ∈ V | |
15 | 13, 14 | f1osn 6873 | . 2 ⊢ {〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} |
16 | 6, 12, 15 | vtocl2g 3558 | 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 1534 ∈ wcel 2099 {csn 4624 〈cop 4630 –1-1-onto→wf1o 6541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-mo 2529 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 |
This theorem is referenced by: f1sng 6875 f1oprswap 6877 f1oprg 6878 f1o2sn 7145 fsnunf 7188 fsnex 7286 suppsnop 8174 mapsnd 8894 ralxpmap 8904 en2sn 9055 en2snOLD 9056 enfixsn 9095 fseqenlem1 10033 canthp1lem2 10662 sumsnf 15707 prodsn 15924 prodsnf 15926 vdwlem8 16942 gsumws1 18775 symg1bas 19329 dprdsn 19977 eupthp1 30000 s1f1 32635 poimirlem16 37031 poimirlem17 37032 poimirlem19 37034 poimirlem20 37035 metakunt25 41588 mapfzcons 42048 sumsnd 44301 1hegrlfgr 47107 |
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