Theorem qseq 39271

Description: The quotient set equal to a class.

This theorem is used when a class 𝐴 is identified with a quotient (dom 𝑅 / 𝑅). In such a situation, every element 𝑢 ∈ 𝐴 is an 𝑅-coset [𝑣]𝑅 for some 𝑣 ∈ dom 𝑅, but there is no requirement that the "witness" 𝑣 be equal to its own block [𝑣]𝑅. 𝐴 is a set of blocks (equivalence classes), not a set of raw witnesses. In particular, when (dom 𝑅 / 𝑅) = 𝐴 is read together with a partition hypothesis 𝑅 Part 𝐴 (defined as dfpart2 39410), 𝐴 is being treated as the set of blocks [𝑣]𝑅; it does not assert any fixed-point condition 𝑣 = [𝑣]𝑅 such as would arise from the mistaken reading 𝑢 ∈ 𝐴 ↔ 𝑢 = [𝑢]𝑅. Cf. dmqsblocks 39505. (Contributed by Peter Mazsa, 19-Oct-2018.)

Assertion

Ref	Expression
qseq	⊢ ((𝐵 / 𝑅) = 𝐴 ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐵,𝑣   𝑢,𝑅,𝑣

Allowed substitution hint:   𝐴(𝑣)

Proof of Theorem qseq

Step	Hyp	Ref	Expression
1		df-qs 8699	. . 3 ⊢ (𝐵 / 𝑅) = {𝑢 ∣ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅}
2	1	eqeq2i 2782	. 2 ⊢ (𝐴 = (𝐵 / 𝑅) ↔ 𝐴 = {𝑢 ∣ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅})
3		eqcom 2776	. 2 ⊢ (𝐴 = (𝐵 / 𝑅) ↔ (𝐵 / 𝑅) = 𝐴)
4		eqabb 2908	. 2 ⊢ (𝐴 = {𝑢 ∣ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅} ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅))
5	2, 3, 4	3bitr3i 304	1 ⊢ ((𝐵 / 𝑅) = 𝐴 ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅))

Syntax hints:   ↔ wb 209  ∀wal 1565   = wceq 1567   ∈ wcel 2149  {cab 2747  ∃wrex 3095  [cec 8691   / cqs 8692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-qs 8699

This theorem is referenced by:  dmqsblocks  39505