Theorem qsalrel 41368

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 41365	. 2 ⊢ (𝐴 / ∼ ) = ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ )
2		qsalrel.2	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝐴)
3	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∼ Er 𝐴)
4		qsalrel.1	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∼ 𝑦)
5	4	ralrimivva 3200	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦)
6	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦)
7		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
8		breq1 5151	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑥 ∼ 𝑦 ↔ 𝑎 ∼ 𝑦))
9	8	ralbidv 3177	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦))
10	9	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = 𝑎) → (∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦))
11	7, 10	rspcdv 3604	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦 → ∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦))
12		qsalrel.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ 𝐴)
13		breq2 5152	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑁 → (𝑎 ∼ 𝑦 ↔ 𝑎 ∼ 𝑁))
14	13	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = 𝑁) → (𝑎 ∼ 𝑦 ↔ 𝑎 ∼ 𝑁))
15	12, 14	rspcdv 3604	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦 → 𝑎 ∼ 𝑁))
16	15	adantr 481	. . . . . . . 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 8756	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → [𝑎] ∼ = [𝑁] ∼ )
20	19	mpteq2dva 5248	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ))
21	20	rneqd 5937	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = ran (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ))
22		eqid 2732	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ) = (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ )
23	12	ne0d 4335	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
24	22, 23	rnmptc 7210	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ) = {[𝑁] ∼ })
25	2	ecss 8751	. . . . 5 ⊢ (𝜑 → [𝑁] ∼ ⊆ 𝐴)
26	3, 18	ersym 8717	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 ∼ 𝑎)
27	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 ∈ 𝐴)
28		elecg 8748	. . . . . . 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 256	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ [𝑁] ∼ )
31	25, 30	eqelssd 4003	. . . 4 ⊢ (𝜑 → [𝑁] ∼ = 𝐴)
32	31	sneqd 4640	. . 3 ⊢ (𝜑 → {[𝑁] ∼ } = {𝐴})
33	21, 24, 32	3eqtrd 2776	. 2 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = {𝐴})
34	1, 33	eqtrid 2784	1 ⊢ (𝜑 → (𝐴 / ∼ ) = {𝐴})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  {csn 4628   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677   Er wer 8702  [cec 8703   / cqs 8704

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-er 8705  df-ec 8707  df-qs 8711

This theorem is referenced by:  0prjspn  41672