Theorem qsxpid 31460

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 484	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = [𝑥](𝐴 × 𝐴))
2		ecxpid 31458	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → [𝑥](𝐴 × 𝐴) = 𝐴)
3	2	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → [𝑥](𝐴 × 𝐴) = 𝐴)
4	1, 3	eqtrd 2778	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
5	4	rexlimiva 3209	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) → 𝑦 = 𝐴)
6	5	adantl 481	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
7		n0 4277	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
8	7	biimpi 215	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
9		simpl 482	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 = 𝐴)
10	2	adantl 481	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑥](𝐴 × 𝐴) = 𝐴)
11	9, 10	eqtr4d 2781	. . . . . . . . 9 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 = [𝑥](𝐴 × 𝐴))
12	11	ex 412	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑦 = [𝑥](𝐴 × 𝐴)))
13	12	ancld 550	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴))))
14	13	eximdv 1921	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴))))
15	8, 14	mpan9 506	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝑦 = 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)))
16		df-rex 3069	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)))
17	15, 16	sylibr 233	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝑦 = 𝐴) → ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴))
18	6, 17	impbida 797	. . 3 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) ↔ 𝑦 = 𝐴))
19		vex 3426	. . . 4 ⊢ 𝑦 ∈ V
20	19	elqs 8516	. . 3 ⊢ (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴))
21		velsn 4574	. . 3 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
22	18, 20, 21	3bitr4g 313	. 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ 𝑦 ∈ {𝐴}))
23	22	eqrdv 2736	1 ⊢ (𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064  ∅c0 4253  {csn 4558   × cxp 5578  [cec 8454   / cqs 8455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ec 8458  df-qs 8462

This theorem is referenced by:  qustriv  31462