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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 6648 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) | |
2 | f1of1 6607 | . . 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 4733 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ⊆ 𝑊) | |
5 | 4 | adantl 482 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐵} ⊆ 𝑊) |
6 | f1ss 6573 | . 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 396 ∈ wcel 2105 ⊆ wss 3933 {csn 4557 〈cop 4563 –1-1→wf1 6345 –1-1-onto→wf1o 6347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 |
This theorem is referenced by: fsnd 6650 uspgr1e 26953 0wlkons1 27827 |
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