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Mirrors > Home > MPE Home > Th. List > xpsng | Structured version Visualization version GIF version |
Description: The Cartesian product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
xpsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstg 6232 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ({𝐴} × {𝐵}):{𝐴}⟶{𝐵}) | |
2 | 1 | adantl 467 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}):{𝐴}⟶{𝐵}) |
3 | fsng 6547 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} × {𝐵}):{𝐴}⟶{𝐵} ↔ ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉})) | |
4 | 2, 3 | mpbid 222 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 {csn 4316 〈cop 4322 × cxp 5247 ⟶wf 6027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 |
This theorem is referenced by: xpsn 6550 f1o2sn 6551 residpr 6552 fmptsn 6577 mpt2sn 7419 repsw1 13739 s1co 13788 xpscg 16426 xpsc0 16428 xpsc1 16429 intopsn 17461 grp1inv 17731 psgnsn 18147 ixpsnbasval 19424 mat1dimelbas 20495 mat1dimscm 20499 mat1dimmul 20500 mat1f1o 20502 m1detdiag 20621 pt1hmeo 21830 nosupbnd2lem1 32198 rngosn3 34055 fmptsnxp 39869 xpprsng 42638 lmod1zr 42810 |
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