Theorem qsresid 38313

Description: Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.)

Assertion

Ref	Expression
qsresid	⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅)

Proof of Theorem qsresid

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elecreseq 8720	. . . . 5 ⊢ (𝑣 ∈ 𝐴 → [𝑣](𝑅 ↾ 𝐴) = [𝑣]𝑅)
2	1	eqeq2d 2740	. . . 4 ⊢ (𝑣 ∈ 𝐴 → (𝑢 = [𝑣](𝑅 ↾ 𝐴) ↔ 𝑢 = [𝑣]𝑅))
3	2	rexbiia 3074	. . 3 ⊢ (∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴) ↔ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅)
4	3	abbii 2796	. 2 ⊢ {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴)} = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅}
5		df-qs 8677	. 2 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴)}
6		df-qs 8677	. 2 ⊢ (𝐴 / 𝑅) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅}
7	4, 5, 6	3eqtr4i 2762	1 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅)

Syntax hints:   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ↾ cres 5640  [cec 8669   / cqs 8670

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 5251  ax-nul 5261  ax-pr 5387

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8673  df-qs 8677

This theorem is referenced by:  n0elim  38642