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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 5154 | . 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
6 | 5 | mptru 1543 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ⊤wtru 1537 ↦ cmpt 5149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-opab 5132 df-mpt 5150 |
This theorem is referenced by: evlsval 20302 madufval 21249 cdj3lem3 30218 cdj3lem3b 30220 partfun 30424 esumsnf 31327 esumrnmpt2 31331 measinb2 31486 eulerpart 31644 fiblem 31660 hoidmvlelem4 42887 smflimsup 43109 |
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