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Mirrors > Home > ILE Home > Th. List > mpompt | 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 4646 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) | |
2 | mpteq1 4048 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) |
4 | mpompt.1 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝐶 = 𝐷) | |
5 | 4 | mpomptx 5912 | . 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
6 | 3, 5 | eqtr3i 2180 | 1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 {csn 3560 〈cop 3563 ∪ ciun 3849 ↦ cmpt 4025 × cxp 4584 ∈ cmpo 5826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-iun 3851 df-opab 4026 df-mpt 4027 df-xp 4592 df-rel 4593 df-oprab 5828 df-mpo 5829 |
This theorem is referenced by: fconstmpo 5916 fnovim 5929 fmpoco 6163 xpf1o 6789 txbas 12669 cnmpt1st 12699 cnmpt2nd 12700 cnmpt2c 12701 cnmpt2t 12704 txhmeo 12730 txswaphmeolem 12731 |
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