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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 5749 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) | |
2 | 1 | mpteq1i 5245 | . 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) |
3 | mpompt.1 | . . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷) | |
4 | 3 | mpomptx 7521 | . 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
5 | 2, 4 | eqtr3i 2763 | 1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 {csn 4629 ⟨cop 4635 ∪ ciun 4998 ↦ cmpt 5232 × cxp 5675 ∈ cmpo 7411 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-iun 5000 df-opab 5212 df-mpt 5233 df-xp 5683 df-rel 5684 df-oprab 7413 df-mpo 7414 |
This theorem is referenced by: fconstmpo 7525 fnov 7540 fmpoco 8081 fimaproj 8121 xpf1o 9139 resfval2 17843 catcisolem 18060 xpccatid 18140 curf2ndf 18200 evlslem4 21637 mdetunilem9 22122 txbas 23071 cnmpt1st 23172 cnmpt2nd 23173 cnmpt2c 23174 cnmpt2t 23177 txhmeo 23307 txswaphmeolem 23308 ptuncnv 23311 ptunhmeo 23312 xpstopnlem1 23313 xkohmeo 23319 prdstmdd 23628 ucnimalem 23785 fmucndlem 23796 fsum2cn 24387 curfv 36468 aks6d1c2p1 40956 fmpocos 41056 idfusubc0 46639 lmod1zr 47174 2arymaptf 47338 |
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