Theorem qsalrel 42608

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 8652	. 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 3181	. . . . . . . 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 5103	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑥 ∼ 𝑦 ↔ 𝑎 ∼ 𝑦))
9	8	ralbidv 3161	. . . . . . . . . 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 3570	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ∼ 𝑦 → ∀𝑦 ∈ 𝐴 𝑎 ∼ 𝑦))
12		qsalrel.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ 𝐴)
13		breq2 5104	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑁 → (𝑎 ∼ 𝑦 ↔ 𝑎 ∼ 𝑁))
14	13	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = 𝑁) → (𝑎 ∼ 𝑦 ↔ 𝑎 ∼ 𝑁))
15	12, 14	rspcdv 3570	. . . . . . . . 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 8702	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → [𝑎] ∼ = [𝑁] ∼ )
20	19	mpteq2dva 5193	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ))
21	20	rneqd 5895	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑎] ∼ ) = ran (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ))
22		eqid 2737	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ) = (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ )
23	12	ne0d 4296	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
24	22, 23	rnmptc 7163	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ [𝑁] ∼ ) = {[𝑁] ∼ })
25	2	ecss 8697	. . . . 5 ⊢ (𝜑 → [𝑁] ∼ ⊆ 𝐴)
26	3, 18	ersym 8658	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 ∼ 𝑎)
27	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 ∈ 𝐴)
28		elecg 8690	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑁 ∈ 𝐴) → (𝑎 ∈ [𝑁] ∼ ↔ 𝑁 ∼ 𝑎))
29	7, 27, 28	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 ∈ [𝑁] ∼ ↔ 𝑁 ∼ 𝑎))
30	26, 29	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ [𝑁] ∼ )
31	25, 30	eqelssd 3957	. . . 4 ⊢ (𝜑 → [𝑁] ∼ = 𝐴)
32	31	sneqd 4594	. . 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {csn 4582   class class class wbr 5100   ↦ cmpt 5181  ran crn 5633   Er wer 8642  [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  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-er 8645  df-ec 8647  df-qs 8651

This theorem is referenced by:  0prjspn  42983