Theorem qsalrel 43260

Description: The quotient set is equal to the singleton of 𝐴 when all elements are related and 𝐴 is nonempty. (Contributed by SN, 8-Jun-2023.)

Hypotheses

Ref	Expression
qsalrel.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∼ 𝑦)
qsalrel.2	⊢ (𝜑 → ∼ Er 𝐴)
qsalrel.3	⊢ (𝜑 → 𝑁 ∈ 𝐴)

Assertion

Ref	Expression
qsalrel	⊢ (𝜑 → (𝐴 / ∼ ) = {𝐴})

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑥, ∼ ,𝑦   𝑦,𝑁

Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem qsalrel

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfqs2 8708	. 2 ⊢ (𝐴 / ∼ ) = ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ )
2		qsalrel.2	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝐴)
3	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∼ Er 𝐴)
4		qsalrel.1	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∼ 𝑦)
5	4	ralrimivva 3206	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦)
6	5	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦)
7		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
8		breq1 5106	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑥 ∼ 𝑦 ↔ 𝑎 ∼ 𝑦))
9	8	ralbidv 3186	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦))
10	9	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = 𝑎) → (∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦))
11	7, 10	rspcdv 3569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦 → ∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦))
12		qsalrel.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ 𝐴)
13		breq2 5107	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑁 → (𝑎 ∼ 𝑦 ↔ 𝑎 ∼ 𝑁))
14	13	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = 𝑁) → (𝑎 ∼ 𝑦 ↔ 𝑎 ∼ 𝑁))
15	12, 14	rspcdv 3569	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦 → 𝑎 ∼ 𝑁))
16	15	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦 → 𝑎 ∼ 𝑁))
17	11, 16	syld 48	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦 → 𝑎 ∼ 𝑁))
18	6, 17	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∼ 𝑁)
19	3, 18	erthi 8758	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → [𝑎] ∼ = [𝑁] ∼ )
20	19	mpteq2dva 5198	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ))
21	20	rneqd 5920	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = ran (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ))
22		eqid 2761	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ) = (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ )
23	12	ne0d 4288	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
24	22, 23	rnmptc 7205	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ) = {[𝑁] ∼ })
25	2	ecss 8753	. . . . 5 ⊢ (𝜑 → [𝑁] ∼ ⊆ 𝐴)
26	3, 18	ersym 8714	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 ∼ 𝑎)
27	12	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 ∈ 𝐴)
28		elecg 8746	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑁 ∈ 𝐴) → (𝑎 ∈ [𝑁] ∼ ↔ 𝑁 ∼ 𝑎))
29	7, 27, 28	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 ∈ [𝑁] ∼ ↔ 𝑁 ∼ 𝑎))
30	26, 29	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ [𝑁] ∼ )
31	25, 30	eqelssd 3952	. . . 4 ⊢ (𝜑 → [𝑁] ∼ = 𝐴)
32	31	sneqd 4596	. . 3 ⊢ (𝜑 → {[𝑁] ∼ } = {𝐴})
33	21, 24, 32	3eqtrd 2800	. 2 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = {𝐴})
34	1, 33	eqtrid 2808	1 ⊢ (𝜑 → (𝐴 / ∼ ) = {𝐴})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {csn 4584   class class class wbr 5103   ↦ cmpt 5186  ran crn 5652   Er wer 8698  [cec 8699   / cqs 8700

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-mpt 5187  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-er 8701  df-ec 8703  df-qs 8707

This theorem is used by:  0prjspn  43618