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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfpart2 | Structured version Visualization version GIF version |
Description: Alternate definition of the partition predicate. (Contributed by Peter Mazsa, 5-Sep-2021.) |
Ref | Expression |
---|---|
dfpart2 | ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-part 38148 | . 2 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴)) | |
2 | df-dmqs 38021 | . . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴) | |
3 | 2 | anbi2i 622 | . 2 ⊢ (( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴) ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) |
4 | 1, 3 | bitri 275 | 1 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 dom cdm 5669 / cqs 8701 DomainQs wdmqs 37579 Disj wdisjALTV 37589 Part wpart 37594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 396 df-dmqs 38021 df-part 38148 |
This theorem is referenced by: parteq1 38156 parteq2 38157 partim 38190 pet0 38197 petid 38199 petidres 38201 petinidres 38203 petxrnidres 38205 petincnvepres 38231 pet 38233 |
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