Theorem qsresid 37706

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		ecres2 37659	. . . . 5 ⊢ (𝑣 ∈ 𝐴 → [𝑣](𝑅 ↾ 𝐴) = [𝑣]𝑅)
2	1	eqeq2d 2737	. . . 4 ⊢ (𝑣 ∈ 𝐴 → (𝑢 = [𝑣](𝑅 ↾ 𝐴) ↔ 𝑢 = [𝑣]𝑅))
3	2	rexbiia 3086	. . 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 8708	. 2 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴)}
6		df-qs 8708	. 2 ⊢ (𝐴 / 𝑅) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅}
7	4, 5, 6	3eqtr4i 2764	1 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅)

Syntax hints:   = wceq 1533   ∈ wcel 2098  {cab 2703  ∃wrex 3064   ↾ cres 5671  [cec 8700   / cqs 8701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8704  df-qs 8708

This theorem is referenced by:  n0elim  38032