![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcnvrefrel4 | Structured version Visualization version GIF version |
Description: Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.) |
Ref | Expression |
---|---|
dfcnvrefrel4 | ⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnvrefrel 37303 | . 2 ⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) | |
2 | cnvref4 37125 | . 2 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ I )) | |
3 | 1, 2 | bianim 37002 | 1 ⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∩ cin 3945 ⊆ wss 3946 I cid 5569 × cxp 5670 dom cdm 5672 ran crn 5673 Rel wrel 5677 CnvRefRel wcnvrefrel 36958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-cnvrefrel 37303 |
This theorem is referenced by: dfcnvrefrel5 37309 dfantisymrel4 37537 |
Copyright terms: Public domain | W3C validator |