Theorem ecex2 38351

Description: Condition for a coset to be a set. (Contributed by Peter Mazsa, 4-May-2019.)

Assertion

Ref	Expression
ecex2	⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ V))

Proof of Theorem ecex2

Step	Hyp	Ref	Expression
1		ecexg 8728	. 2 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → [𝐵](𝑅 ↾ 𝐴) ∈ V)
2		ecres2 38302	. . 3 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)
3	2	eleq1d 2820	. 2 ⊢ (𝐵 ∈ 𝐴 → ([𝐵](𝑅 ↾ 𝐴) ∈ V ↔ [𝐵]𝑅 ∈ V))
4	1, 3	syl5ibcom 245	1 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ V))

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3464   ↾ cres 5661  [cec 8722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8726

This theorem is referenced by:  uniqsALTV  38352