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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 5758 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) | |
| 2 | 1 | mpteq1i 5238 | . 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) |
| 3 | mpompt.1 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝐶 = 𝐷) | |
| 4 | 3 | mpomptx 7546 | . 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 1540 {csn 4626 〈cop 4632 ∪ ciun 4991 ↦ cmpt 5225 × cxp 5683 ∈ cmpo 7433 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-iun 4993 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: fconstmpo 7550 fnov 7564 fmpoco 8120 fimaproj 8160 xpf1o 9179 resfval2 17938 idfusubc0 17944 catcisolem 18155 xpccatid 18233 curf2ndf 18292 evlslem4 22100 mdetunilem9 22626 txbas 23575 cnmpt1st 23676 cnmpt2nd 23677 cnmpt2c 23678 cnmpt2t 23681 txhmeo 23811 txswaphmeolem 23812 ptuncnv 23815 ptunhmeo 23816 xpstopnlem1 23817 xkohmeo 23823 prdstmdd 24132 ucnimalem 24289 fmucndlem 24300 fsum2cn 24895 elrgspnlem2 33247 curfv 37607 aks6d1c2p1 42119 aks6d1c3 42124 aks6d1c4 42125 aks6d1c6lem2 42172 aks6d1c6lem4 42174 aks6d1c7lem1 42181 fmpocos 42275 lmod1zr 48410 2arymaptf 48573 |
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