![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > f1sng | Structured version Visualization version GIF version |
Description: A singleton of an ordered pair is a one-to-one function. (Contributed by AV, 17-Apr-2021.) |
Ref | Expression |
---|---|
f1sng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1osng 6890 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) | |
2 | f1of1 6848 | . . 3 ⊢ ({〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} → {〈𝐴, 𝐵〉}:{𝐴}–1-1→{𝐵}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→{𝐵}) |
4 | snssi 4813 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ⊆ 𝑊) | |
5 | 4 | adantl 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐵} ⊆ 𝑊) |
6 | f1ss 6810 | . 2 ⊢ (({〈𝐴, 𝐵〉}:{𝐴}–1-1→{𝐵} ∧ {𝐵} ⊆ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) | |
7 | 3, 5, 6 | syl2anc 584 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 〈cop 4637 –1-1→wf1 6560 –1-1-onto→wf1o 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: fsnd 6892 uspgr1e 29276 0wlkons1 30150 f1sn2g 48681 |
Copyright terms: Public domain | W3C validator |