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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcnvrefrel5 | Structured version Visualization version GIF version |
Description: Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.) |
Ref | Expression |
---|---|
dfcnvrefrel5 | ⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcnvrefrel4 37271 | . 2 ⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅)) | |
2 | cnvref5 37089 | . 2 ⊢ (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦))) | |
3 | 1, 2 | bianim 36965 | 1 ⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ⊆ wss 3945 class class class wbr 5142 I cid 5567 Rel wrel 5675 CnvRefRel wcnvrefrel 36921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-dm 5680 df-rn 5681 df-res 5682 df-cnvrefrel 37266 |
This theorem is referenced by: dfantisymrel5 37501 |
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