Theorem qseq 38984

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 39123), 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 39218. (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 8651	. . 3 ⊢ (𝐵 / 𝑅) = {𝑢 ∣ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅}
2	1	eqeq2i 2750	. 2 ⊢ (𝐴 = (𝐵 / 𝑅) ↔ 𝐴 = {𝑢 ∣ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅})
3		eqcom 2744	. 2 ⊢ (𝐴 = (𝐵 / 𝑅) ↔ (𝐵 / 𝑅) = 𝐴)
4		eqabb 2876	. 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 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  [cec 8643   / cqs 8644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-qs 8651

This theorem is referenced by:  dmqsblocks  39218