Theorem qsresid 38503

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 8687	. . . . 5 ⊢ (𝑣 ∈ 𝐴 → [𝑣](𝑅 ↾ 𝐴) = [𝑣]𝑅)
2	1	eqeq2d 2748	. . . 4 ⊢ (𝑣 ∈ 𝐴 → (𝑢 = [𝑣](𝑅 ↾ 𝐴) ↔ 𝑢 = [𝑣]𝑅))
3	2	rexbiia 3082	. . 3 ⊢ (∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴) ↔ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅)
4	3	abbii 2804	. 2 ⊢ {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴)} = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅}
5		df-qs 8643	. 2 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴)}
6		df-qs 8643	. 2 ⊢ (𝐴 / 𝑅) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅}
7	4, 5, 6	3eqtr4i 2770	1 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅)

Syntax hints:   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3061   ↾ cres 5627  [cec 8635   / cqs 8636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639  df-qs 8643

This theorem is referenced by:  n0elim  38907