Theorem qsalrel 42279

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 42276	. 2 ⊢ (𝐴 / ∼ ) = ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ )
2		qsalrel.2	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝐴)
3	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∼ Er 𝐴)
4		qsalrel.1	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∼ 𝑦)
5	4	ralrimivva 3175	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦)
6	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦)
7		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
8		breq1 5094	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑥 ∼ 𝑦 ↔ 𝑎 ∼ 𝑦))
9	8	ralbidv 3155	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦))
10	9	adantl 481	. . . . . . . . 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 5095	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑁 → (𝑎 ∼ 𝑦 ↔ 𝑎 ∼ 𝑁))
14	13	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = 𝑁) → (𝑎 ∼ 𝑦 ↔ 𝑎 ∼ 𝑁))
15	12, 14	rspcdv 3569	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦 → 𝑎 ∼ 𝑁))
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦 → 𝑎 ∼ 𝑁))
17	11, 16	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦 → 𝑎 ∼ 𝑁))
18	6, 17	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∼ 𝑁)
19	3, 18	erthi 8678	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → [𝑎] ∼ = [𝑁] ∼ )
20	19	mpteq2dva 5184	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ))
21	20	rneqd 5878	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = ran (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ))
22		eqid 2731	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ) = (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ )
23	12	ne0d 4292	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
24	22, 23	rnmptc 7141	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ) = {[𝑁] ∼ })
25	2	ecss 8673	. . . . 5 ⊢ (𝜑 → [𝑁] ∼ ⊆ 𝐴)
26	3, 18	ersym 8634	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 ∼ 𝑎)
27	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 ∈ 𝐴)
28		elecg 8666	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑁 ∈ 𝐴) → (𝑎 ∈ [𝑁] ∼ ↔ 𝑁 ∼ 𝑎))
29	7, 27, 28	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 ∈ [𝑁] ∼ ↔ 𝑁 ∼ 𝑎))
30	26, 29	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ [𝑁] ∼ )
31	25, 30	eqelssd 3956	. . . 4 ⊢ (𝜑 → [𝑁] ∼ = 𝐴)
32	31	sneqd 4588	. . 3 ⊢ (𝜑 → {[𝑁] ∼ } = {𝐴})
33	21, 24, 32	3eqtrd 2770	. 2 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = {𝐴})
34	1, 33	eqtrid 2778	1 ⊢ (𝜑 → (𝐴 / ∼ ) = {𝐴})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {csn 4576   class class class wbr 5091   ↦ cmpt 5172  ran crn 5617   Er wer 8619  [cec 8620   / cqs 8621

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  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-mpt 5173  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-er 8622  df-ec 8624  df-qs 8628

This theorem is referenced by:  0prjspn  42667