Theorem parteq2 38731

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 2752	. . 3 ⊢ (𝐴 = 𝐵 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐵))
2	1	anbi2d 629	. 2 ⊢ (𝐴 = 𝐵 → (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐵)))
3		dfpart2 38725	. 2 ⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
4		dfpart2 38725	. 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 206   ∧ wa 395   = wceq 1537  dom cdm 5700   / cqs 8762   Disj wdisjALTV 38169   Part wpart 38174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-dmqs 38595  df-part 38722

This theorem is referenced by:  parteq12  38732