Theorem qseq 38635

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 38756), 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 38840. (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 8679	. . 3 ⊢ (𝐵 / 𝑅) = {𝑢 ∣ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅}
2	1	eqeq2i 2743	. 2 ⊢ (𝐴 = (𝐵 / 𝑅) ↔ 𝐴 = {𝑢 ∣ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅})
3		eqcom 2737	. 2 ⊢ (𝐴 = (𝐵 / 𝑅) ↔ (𝐵 / 𝑅) = 𝐴)
4		eqabb 2868	. 2 ⊢ (𝐴 = {𝑢 ∣ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅} ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅))
5	2, 3, 4	3bitr3i 301	1 ⊢ ((𝐵 / 𝑅) = 𝐴 ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅))

Syntax hints:   ↔ wb 206  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {cab 2708  ∃wrex 3054  [cec 8671   / cqs 8672

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 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-qs 8679

This theorem is referenced by:  dmqsblocks  38840