Theorem qseq 38686

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 38807), 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 38891. (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 8623	. . 3 ⊢ (𝐵 / 𝑅) = {𝑢 ∣ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅}
2	1	eqeq2i 2744	. 2 ⊢ (𝐴 = (𝐵 / 𝑅) ↔ 𝐴 = {𝑢 ∣ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅})
3		eqcom 2738	. 2 ⊢ (𝐴 = (𝐵 / 𝑅) ↔ (𝐵 / 𝑅) = 𝐴)
4		eqabb 2870	. 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 1539   = wceq 1541   ∈ wcel 2111  {cab 2709  ∃wrex 3056  [cec 8615   / cqs 8616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-qs 8623

This theorem is referenced by:  dmqsblocks  38891