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Mirrors > Home > MPE Home > Th. List > fconstmpo | Structured version Visualization version GIF version |
Description: Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.) |
Ref | Expression |
---|---|
fconstmpo | ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstmpt 5620 | . 2 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) | |
2 | eqidd 2739 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝐶 = 𝐶) | |
3 | 2 | mpompt 7333 | . 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
4 | 1, 3 | eqtri 2766 | 1 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 {csn 4550 〈cop 4556 ↦ cmpt 5144 × cxp 5558 ∈ cmpo 7224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pr 5331 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-sn 4551 df-pr 4553 df-op 4557 df-iun 4915 df-opab 5125 df-mpt 5145 df-xp 5566 df-rel 5567 df-oprab 7226 df-mpo 7227 |
This theorem is referenced by: tposconst 8015 mat0op 21329 matsc 21360 mdetrsca2 21514 smadiadetglem2 21582 fedgmullem2 31438 |
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