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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 6089 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) | |
2 | f1of1 6049 | . . 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 4280 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ⊆ 𝑊) | |
5 | 4 | adantl 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐵} ⊆ 𝑊) |
6 | f1ss 6019 | . 2 ⊢ (({〈𝐴, 𝐵〉}:{𝐴}–1-1→{𝐵} ∧ {𝐵} ⊆ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) | |
7 | 3, 5, 6 | syl2anc 691 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 〈cop 4131 –1-1→wf1 5801 –1-1-onto→wf1o 5803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: fsnd 6091 uspgr1e 40470 0wlkOns1 41289 |
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