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Mirrors > Home > ILE Home > Th. List > mpteq2i | GIF version |
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpteq2i.1 | ⊢ 𝐵 = 𝐶 |
Ref | Expression |
---|---|
mpteq2i | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq2i.1 | . . 3 ⊢ 𝐵 = 𝐶 | |
2 | 1 | a1i 9 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) |
3 | 2 | mpteq2ia 4103 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ∈ wcel 2159 ↦ cmpt 4078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-11 1516 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2170 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2175 df-cleq 2181 df-clel 2184 df-ral 2472 df-opab 4079 df-mpt 4080 |
This theorem is referenced by: frecsuc 6425 fodjuomni 7164 fodjumkv 7175 axcaucvg 7916 0tonninf 10456 1tonninf 10457 cbvsum 11385 cbvprod 11583 eirraplem 11801 cnmpt12f 14169 fsumcncntop 14439 dvef 14571 nninfsellemqall 15148 nninfomni 15152 exmidsbthr 15155 |
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