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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 35602 | . 2 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅) | |
2 | dfres3 5857 | . . 3 ⊢ (𝑅 ↾ dom 𝑅) = (𝑅 ∩ (dom 𝑅 × ran 𝑅)) | |
3 | 2 | eqeq1i 2826 | . 2 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅) |
4 | 1, 3 | bitri 277 | 1 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∩ cin 3934 × cxp 5552 dom cdm 5554 ran crn 5555 ↾ cres 5556 Rel wrel 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-dm 5564 df-rn 5565 df-res 5566 |
This theorem is referenced by: elrels6 35729 dfrefrel2 35754 dfcnvrefrel2 35767 dfsymrel2 35784 dftrrel2 35812 |
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