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Mirrors > Home > MPE Home > Th. List > cleq1 | Structured version Visualization version GIF version |
Description: Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.) |
Ref | Expression |
---|---|
cleq1 | ⊢ (𝑅 = 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} = ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cleq1lem 14330 | . . 3 ⊢ (𝑅 = 𝑆 → ((𝑅 ⊆ 𝑟 ∧ 𝜑) ↔ (𝑆 ⊆ 𝑟 ∧ 𝜑))) | |
2 | 1 | abbidv 2882 | . 2 ⊢ (𝑅 = 𝑆 → {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} = {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)}) |
3 | 2 | inteqd 4872 | 1 ⊢ (𝑅 = 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} = ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 {cab 2796 ⊆ wss 3933 ∩ cint 4867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-ral 3140 df-in 3940 df-ss 3949 df-int 4868 |
This theorem is referenced by: trcleq1 14337 |
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