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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 6177 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) | |
2 | f1of1 6136 | . . 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 4339 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ⊆ 𝑊) | |
5 | 4 | adantl 482 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐵} ⊆ 𝑊) |
6 | f1ss 6106 | . 2 ⊢ (({〈𝐴, 𝐵〉}:{𝐴}–1-1→{𝐵} ∧ {𝐵} ⊆ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) | |
7 | 3, 5, 6 | syl2anc 693 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ⊆ wss 3574 {csn 4177 〈cop 4183 –1-1→wf1 5885 –1-1-onto→wf1o 5887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 |
This theorem is referenced by: fsnd 6179 uspgr1e 26136 0wlkons1 26982 |
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