Theorem elecex 8768

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 8721	. 2 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → [𝐵](𝑅 ↾ 𝐴) ∈ V)
2		elecreseq 8767	. . 3 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)
3	2	eleq1d 2846	. 2 ⊢ (𝐵 ∈ 𝐴 → ([𝐵](𝑅 ↾ 𝐴) ∈ V ↔ [𝐵]𝑅 ∈ V))
4	1, 3	syl5ibcom 248	1 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ V))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3451   ↾ cres 5653  [cec 8715

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8719

This theorem is used by:  ecelqs  8788  uniqs  8794  disjqmap2  39758