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| Mirrors > Home > MPE Home > Th. List > xpsng | Structured version Visualization version GIF version | ||
| Description: The Cartesian product of two singletons is the singleton consisting in the associated ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
| Ref | Expression |
|---|---|
| xpsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fconstg 6795 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ({𝐴} × {𝐵}):{𝐴}⟶{𝐵}) | |
| 2 | 1 | adantl 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}):{𝐴}⟶{𝐵}) |
| 3 | fsng 7157 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} × {𝐵}):{𝐴}⟶{𝐵} ↔ ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉})) | |
| 4 | 2, 3 | mpbid 232 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 × cxp 5683 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 |
| This theorem is referenced by: xpprsng 7160 xpsn 7161 f1o2sn 7162 residpr 7163 fmptsn 7187 f1ofvswap 7326 mposn 8128 repsw1 14821 s1co 14872 intopsn 18667 grp1inv 19066 psgnsn 19538 ixpsnbasval 21215 mat1dimelbas 22477 mat1dimscm 22481 mat1dimmul 22482 mat1f1o 22484 m1detdiag 22603 pt1hmeo 23814 nosupbnd2lem1 27760 cosnop 32704 cshw1s2 32945 rngosn3 37931 fmptsnxp 45174 lmod1zr 48410 cosn 48745 |
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