Theorem elecex 8698

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

Assertion

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

Proof of Theorem elecex

Step	Hyp	Ref	Expression
1		ecexg 8652	. 2 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → [𝐵](𝑅 ↾ 𝐴) ∈ V)
2		elecreseq 8697	. . 3 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)
3	2	eleq1d 2813	. 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 3444   ↾ cres 5633  [cec 8646

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

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 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8650

This theorem is referenced by:  ecelqs  8718  uniqs  8724