Theorem elecex 8723

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 8677	. 2 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → [𝐵](𝑅 ↾ 𝐴) ∈ V)
2		elecreseq 8722	. . 3 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)
3	2	eleq1d 2814	. 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 3450   ↾ cres 5642  [cec 8671

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 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-xp 5646  df-rel 5647  df-cnv 5648  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-ec 8675

This theorem is referenced by:  ecelqs  8743  uniqs  8749