Theorem qsxpid 31083

Description: The quotient set of a cartesian product is trivial. (Contributed by Thierry Arnoux, 16-Jan-2024.)

Assertion

Ref	Expression
qsxpid	⊢ (𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})

Proof of Theorem qsxpid

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 488	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = [𝑥](𝐴 × 𝐴))
2		ecxpid 31081	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → [𝑥](𝐴 × 𝐴) = 𝐴)
3	2	adantr 484	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → [𝑥](𝐴 × 𝐴) = 𝐴)
4	1, 3	eqtrd 2793	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
5	4	rexlimiva 3205	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) → 𝑦 = 𝐴)
6	5	adantl 485	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
7		n0 4247	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
8	7	biimpi 219	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
9		simpl 486	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 = 𝐴)
10	2	adantl 485	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑥](𝐴 × 𝐴) = 𝐴)
11	9, 10	eqtr4d 2796	. . . . . . . . 9 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 = [𝑥](𝐴 × 𝐴))
12	11	ex 416	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑦 = [𝑥](𝐴 × 𝐴)))
13	12	ancld 554	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴))))
14	13	eximdv 1918	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴))))
15	8, 14	mpan9 510	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝑦 = 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)))
16		df-rex 3076	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)))
17	15, 16	sylibr 237	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝑦 = 𝐴) → ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴))
18	6, 17	impbida 800	. . 3 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) ↔ 𝑦 = 𝐴))
19		vex 3413	. . . 4 ⊢ 𝑦 ∈ V
20	19	elqs 8364	. . 3 ⊢ (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴))
21		velsn 4541	. . 3 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
22	18, 20, 21	3bitr4g 317	. 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ 𝑦 ∈ {𝐴}))
23	22	eqrdv 2756	1 ⊢ (𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071  ∅c0 4227  {csn 4525   × cxp 5525  [cec 8302   / cqs 8303

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-xp 5533  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-ec 8306  df-qs 8310

This theorem is referenced by:  qustriv  31085