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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 14339 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) ↔ (𝑅 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))) | |
2 | 1 | cbvabv 2886 | 1 ⊢ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 {cab 2796 ⊆ wss 3933 ∘ ccom 5552 |
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-nfc 2960 df-in 3940 df-ss 3949 df-br 5058 df-opab 5120 df-co 5557 |
This theorem is referenced by: (None) |
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