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Mirrors > Home > MPE Home > Th. List > cvbtrcl | Structured version Visualization version GIF version |
Description: Change of bound variable in class of all transitive relations which are supersets of a relation. (Contributed by RP, 5-May-2020.) |
Ref | Expression |
---|---|
cvbtrcl | ⊢ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trcleq2lem 14554 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) ↔ (𝑅 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))) | |
2 | 1 | cbvabv 2811 | 1 ⊢ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 {cab 2714 ⊆ wss 3866 ∘ ccom 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-in 3873 df-ss 3883 df-br 5054 df-opab 5116 df-co 5560 |
This theorem is referenced by: (None) |
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