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Mirrors > Home > MPE Home > Th. List > mpteq12i | Structured version Visualization version GIF version |
Description: An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpteq12i.1 | ⊢ 𝐴 = 𝐶 |
mpteq12i.2 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
mpteq12i | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq12i.1 | . . . 4 ⊢ 𝐴 = 𝐶 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 𝐴 = 𝐶) |
3 | mpteq12i.2 | . . . 4 ⊢ 𝐵 = 𝐷 | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 𝐵 = 𝐷) |
5 | 2, 4 | mpteq12dv 5239 | . 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
6 | 5 | mptru 1547 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ⊤wtru 1541 ↦ cmpt 5231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-opab 5211 df-mpt 5232 |
This theorem is referenced by: partfun 6697 evlsval 21869 madufval 22360 cdj3lem3 31959 cdj3lem3b 31961 esumsnf 33361 esumrnmpt2 33365 measinb2 33520 eulerpart 33680 fiblem 33696 hoidmvlelem4 45613 smflimsup 45843 |
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