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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 5578 | . 2 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) | |
2 | eqidd 2799 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝐶 = 𝐶) | |
3 | 2 | mpompt 7245 | . 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
4 | 1, 3 | eqtri 2821 | 1 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 {csn 4525 〈cop 4531 ↦ cmpt 5110 × cxp 5517 ∈ cmpo 7137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-iun 4883 df-opab 5093 df-mpt 5111 df-xp 5525 df-rel 5526 df-oprab 7139 df-mpo 7140 |
This theorem is referenced by: tposconst 7913 mat0op 21024 matsc 21055 mdetrsca2 21209 smadiadetglem2 21277 fedgmullem2 31114 |
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