Theorem qsresid 39372

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 8774	. . . . 5 ⊢ (𝑣 ∈ 𝐴 → [𝑣](𝑅 ↾ 𝐴) = [𝑣]𝑅)
2	1	eqeq2d 2771	. . . 4 ⊢ (𝑣 ∈ 𝐴 → (𝑢 = [𝑣](𝑅 ↾ 𝐴) ↔ 𝑢 = [𝑣]𝑅))
3	2	rexbiia 3107	. . 3 ⊢ (∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴) ↔ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅)
4	3	abbii 2827	. 2 ⊢ {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴)} = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅}
5		df-qs 8730	. 2 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴)}
6		df-qs 8730	. 2 ⊢ (𝐴 / 𝑅) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅}
7	4, 5, 6	3eqtr4i 2793	1 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅)

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086   ↾ cres 5653  [cec 8722   / cqs 8723

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8726  df-qs 8730

This theorem is used by:  n0elim  39776