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Mirrors > Home > MPE Home > Th. List > mpompt | Structured version Visualization version GIF version |
Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
mpompt.1 | ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
mpompt | ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxpconst 5709 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) | |
2 | 1 | mpteq1i 5206 | . 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) |
3 | mpompt.1 | . . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷) | |
4 | 3 | mpomptx 7474 | . 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
5 | 2, 4 | eqtr3i 2767 | 1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 {csn 4591 ⟨cop 4597 ∪ ciun 4959 ↦ cmpt 5193 × cxp 5636 ∈ cmpo 7364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-iun 4961 df-opab 5173 df-mpt 5194 df-xp 5644 df-rel 5645 df-oprab 7366 df-mpo 7367 |
This theorem is referenced by: fconstmpo 7478 fnov 7492 fmpoco 8032 fimaproj 8072 xpf1o 9090 resfval2 17786 catcisolem 18003 xpccatid 18083 curf2ndf 18143 evlslem4 21500 mdetunilem9 21985 txbas 22934 cnmpt1st 23035 cnmpt2nd 23036 cnmpt2c 23037 cnmpt2t 23040 txhmeo 23170 txswaphmeolem 23171 ptuncnv 23174 ptunhmeo 23175 xpstopnlem1 23176 xkohmeo 23182 prdstmdd 23491 ucnimalem 23648 fmucndlem 23659 fsum2cn 24250 curfv 36087 aks6d1c2p1 40577 idfusubc0 46237 lmod1zr 46648 2arymaptf 46812 |
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