Theorem ecunres 38732

Description: The restricted union coset of 𝐵. (Contributed by Peter Mazsa, 28-Jan-2026.)

Assertion

Ref	Expression
ecunres	⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))

Proof of Theorem ecunres

Step	Hyp	Ref	Expression
1		resundir 5954	. . 3 ⊢ ((𝑅 ∪ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴))
2	1	eceq2i 8680	. 2 ⊢ [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = [𝐵]((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴))
3		ecun 38731	. 2 ⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ↾ 𝐴) ∪ (𝑆 ↾ 𝐴)) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))
4	2, 3	eqtrid 2784	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∪ cun 3888   ↾ cres 5627  [cec 8635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639

This theorem is referenced by:  ecuncnvepres  38733