Theorem qsalrel 43386

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 8728	. 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 3205	. . . . . . . 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 5105	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑥 ∼ 𝑦 ↔ 𝑎 ∼ 𝑦))
9	8	ralbidv 3185	. . . . . . . . . 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 3568	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦 → ∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦))
12		qsalrel.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ 𝐴)
13		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑁 → (𝑎 ∼ 𝑦 ↔ 𝑎 ∼ 𝑁))
14	13	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = 𝑁) → (𝑎 ∼ 𝑦 ↔ 𝑎 ∼ 𝑁))
15	12, 14	rspcdv 3568	. . . . . . . . 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 8778	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → [𝑎] ∼ = [𝑁] ∼ )
20	19	mpteq2dva 5197	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ))
21	20	rneqd 5919	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = ran (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ))
22		eqid 2760	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ) = (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ )
23	12	ne0d 4287	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
24	22, 23	rnmptc 7214	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ) = {[𝑁] ∼ })
25	2	ecss 8773	. . . . 5 ⊢ (𝜑 → [𝑁] ∼ ⊆ 𝐴)
26	3, 18	ersym 8734	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 ∼ 𝑎)
27	12	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 ∈ 𝐴)
28		elecg 8766	. . . . . . 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 3929	. . . 4 ⊢ (𝜑 → [𝑁] ∼ = 𝐴)
32	31	sneqd 4592	. . 3 ⊢ (𝜑 → {[𝑁] ∼ } = {𝐴})
33	21, 24, 32	3eqtrd 2799	. 2 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = {𝐴})
34	1, 33	eqtrid 2807	1 ⊢ (𝜑 → (𝐴 / ∼ ) = {𝐴})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {csn 4584   class class class wbr 5102   ↦ cmpt 5185  ran crn 5651   Er wer 8718  [cec 8719   / cqs 8720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-mpt 5186  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-er 8721  df-ec 8723  df-qs 8727

This theorem is used by:  0prjspn  43758