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Mirrors > Home > ILE Home > Th. List > f1sng | 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 5481 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) | |
2 | f1of1 5439 | . . 3 ⊢ ({〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} → {〈𝐴, 𝐵〉}:{𝐴}–1-1→{𝐵}) | |
3 | 1, 2 | syl 14 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→{𝐵}) |
4 | snssi 3722 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ⊆ 𝑊) | |
5 | 4 | adantl 275 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐵} ⊆ 𝑊) |
6 | f1ss 5407 | . 2 ⊢ (({〈𝐴, 𝐵〉}:{𝐴}–1-1→{𝐵} ∧ {𝐵} ⊆ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) | |
7 | 3, 5, 6 | syl2anc 409 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 ⊆ wss 3121 {csn 3581 〈cop 3584 –1-1→wf1 5193 –1-1-onto→wf1o 5195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 |
This theorem is referenced by: fsnd 5483 |
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