![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > mpt2mpt | 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 |
---|---|
mpt2mpt.1 | ⊢ (z = 〈x, y〉 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
mpt2mpt | ⊢ (z ∈ (A × B) ↦ 𝐶) = (x ∈ A, y ∈ B ↦ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxpconst 4343 | . . 3 ⊢ ∪ x ∈ A ({x} × B) = (A × B) | |
2 | mpteq1 3832 | . . 3 ⊢ (∪ x ∈ A ({x} × B) = (A × B) → (z ∈ ∪ x ∈ A ({x} × B) ↦ 𝐶) = (z ∈ (A × B) ↦ 𝐶)) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ (z ∈ ∪ x ∈ A ({x} × B) ↦ 𝐶) = (z ∈ (A × B) ↦ 𝐶) |
4 | mpt2mpt.1 | . . 3 ⊢ (z = 〈x, y〉 → 𝐶 = 𝐷) | |
5 | 4 | mpt2mptx 5537 | . 2 ⊢ (z ∈ ∪ x ∈ A ({x} × B) ↦ 𝐶) = (x ∈ A, y ∈ B ↦ 𝐷) |
6 | 3, 5 | eqtr3i 2059 | 1 ⊢ (z ∈ (A × B) ↦ 𝐶) = (x ∈ A, y ∈ B ↦ 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 {csn 3367 〈cop 3370 ∪ ciun 3648 ↦ cmpt 3809 × cxp 4286 ↦ cmpt2 5457 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-iun 3650 df-opab 3810 df-mpt 3811 df-xp 4294 df-rel 4295 df-oprab 5459 df-mpt2 5460 |
This theorem is referenced by: fnovim 5551 fmpt2co 5779 |
Copyright terms: Public domain | W3C validator |