Theorem parteq2 38156

Description: Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024.)

Assertion

Ref	Expression
parteq2	⊢ (𝐴 = 𝐵 → (𝑅 Part 𝐴 ↔ 𝑅 Part 𝐵))

Proof of Theorem parteq2

Step	Hyp	Ref	Expression
1		eqeq2 2738	. . 3 ⊢ (𝐴 = 𝐵 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐵))
2	1	anbi2d 628	. 2 ⊢ (𝐴 = 𝐵 → (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐵)))
3		dfpart2 38150	. 2 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
4		dfpart2 38150	. 2 ⊢ (𝑅 Part 𝐵 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐵))
5	2, 3, 4	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (𝑅 Part 𝐴 ↔ 𝑅 Part 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  dom cdm 5669   / cqs 8701   Disj wdisjALTV 37588   Part wpart 37593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2718  df-dmqs 38020  df-part 38147

This theorem is referenced by:  parteq12  38157