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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfrel6 | Structured version Visualization version GIF version |
Description: Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.) |
Ref | Expression |
---|---|
dfrel6 | ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrel5 35484 | . 2 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅) | |
2 | dfres3 5851 | . . 3 ⊢ (𝑅 ↾ dom 𝑅) = (𝑅 ∩ (dom 𝑅 × ran 𝑅)) | |
3 | 2 | eqeq1i 2823 | . 2 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅) |
4 | 1, 3 | bitri 276 | 1 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∩ cin 3932 × cxp 5546 dom cdm 5548 ran crn 5549 ↾ cres 5550 Rel wrel 5553 |
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 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 |
This theorem is referenced by: elrels6 35610 dfrefrel2 35635 dfcnvrefrel2 35648 dfsymrel2 35665 dftrrel2 35693 |
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