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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 14344 | . . 3 ⊢ (𝑅 = 𝑆 → ((𝑅 ⊆ 𝑟 ∧ 𝜑) ↔ (𝑆 ⊆ 𝑟 ∧ 𝜑))) | |
2 | 1 | abbidv 2887 | . 2 ⊢ (𝑅 = 𝑆 → {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} = {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)}) |
3 | 2 | inteqd 4883 | 1 ⊢ (𝑅 = 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} = ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 {cab 2801 ⊆ wss 3938 ∩ cint 4878 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-ral 3145 df-in 3945 df-ss 3954 df-int 4879 |
This theorem is referenced by: trcleq1 14351 |
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