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Mirrors > Home > ILE Home > Th. List > mpt2eq123i | GIF version |
Description: An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.) |
Ref | Expression |
---|---|
mpt2eq123i.1 | ⊢ A = 𝐷 |
mpt2eq123i.2 | ⊢ B = 𝐸 |
mpt2eq123i.3 | ⊢ 𝐶 = 𝐹 |
Ref | Expression |
---|---|
mpt2eq123i | ⊢ (x ∈ A, y ∈ B ↦ 𝐶) = (x ∈ 𝐷, y ∈ 𝐸 ↦ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt2eq123i.1 | . . . 4 ⊢ A = 𝐷 | |
2 | 1 | a1i 9 | . . 3 ⊢ ( ⊤ → A = 𝐷) |
3 | mpt2eq123i.2 | . . . 4 ⊢ B = 𝐸 | |
4 | 3 | a1i 9 | . . 3 ⊢ ( ⊤ → B = 𝐸) |
5 | mpt2eq123i.3 | . . . 4 ⊢ 𝐶 = 𝐹 | |
6 | 5 | a1i 9 | . . 3 ⊢ ( ⊤ → 𝐶 = 𝐹) |
7 | 2, 4, 6 | mpt2eq123dv 5509 | . 2 ⊢ ( ⊤ → (x ∈ A, y ∈ B ↦ 𝐶) = (x ∈ 𝐷, y ∈ 𝐸 ↦ 𝐹)) |
8 | 7 | trud 1251 | 1 ⊢ (x ∈ A, y ∈ B ↦ 𝐶) = (x ∈ 𝐷, y ∈ 𝐸 ↦ 𝐹) |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ⊤ wtru 1243 ↦ 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-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-oprab 5459 df-mpt2 5460 |
This theorem is referenced by: ofmres 5705 |
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