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Theorem parteq2 38247
Description: Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024.)
Assertion
Ref Expression
parteq2 (𝐴 = 𝐵 → (𝑅 Part 𝐴𝑅 Part 𝐵))

Proof of Theorem parteq2
StepHypRef Expression
1 eqeq2 2740 . . 3 (𝐴 = 𝐵 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐵))
21anbi2d 629 . 2 (𝐴 = 𝐵 → (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐵)))
3 dfpart2 38241 . 2 (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
4 dfpart2 38241 . 2 (𝑅 Part 𝐵 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐵))
52, 3, 43bitr4g 314 1 (𝐴 = 𝐵 → (𝑅 Part 𝐴𝑅 Part 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  dom cdm 5678   / cqs 8723   Disj wdisjALTV 37682   Part wpart 37687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-cleq 2720  df-dmqs 38111  df-part 38238
This theorem is referenced by:  parteq12  38248
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