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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 5115 | . 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
6 | 5 | mptru 1545 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ⊤wtru 1539 ↦ cmpt 5110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-opab 5093 df-mpt 5111 |
This theorem is referenced by: partfun 6467 evlsval 20758 madufval 21242 cdj3lem3 30221 cdj3lem3b 30223 esumsnf 31433 esumrnmpt2 31437 measinb2 31592 eulerpart 31750 fiblem 31766 hoidmvlelem4 43237 smflimsup 43459 |
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