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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 5607 | . 2 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) | |
2 | eqidd 2821 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝐶 = 𝐶) | |
3 | 2 | mpompt 7259 | . 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
4 | 1, 3 | eqtri 2843 | 1 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 {csn 4560 〈cop 4566 ↦ cmpt 5139 × cxp 5546 ∈ cmpo 7151 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-iun 4914 df-opab 5122 df-mpt 5140 df-xp 5554 df-rel 5555 df-oprab 7153 df-mpo 7154 |
This theorem is referenced by: tposconst 7923 mat0op 21021 matsc 21052 mdetrsca2 21206 smadiadetglem2 21274 fedgmullem2 31048 |
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