Theorem qsxpid 32150

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 485	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = [𝑥](𝐴 × 𝐴))
2		ecxpid 32148	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → [𝑥](𝐴 × 𝐴) = 𝐴)
3	2	adantr 481	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → [𝑥](𝐴 × 𝐴) = 𝐴)
4	1, 3	eqtrd 2776	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
5	4	rexlimiva 3144	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) → 𝑦 = 𝐴)
6	5	adantl 482	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
7		n0 4306	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
8	7	biimpi 215	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
9		simpl 483	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 = 𝐴)
10	2	adantl 482	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑥](𝐴 × 𝐴) = 𝐴)
11	9, 10	eqtr4d 2779	. . . . . . . . 9 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 = [𝑥](𝐴 × 𝐴))
12	11	ex 413	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑦 = [𝑥](𝐴 × 𝐴)))
13	12	ancld 551	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴))))
14	13	eximdv 1920	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴))))
15	8, 14	mpan9 507	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝑦 = 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)))
16		df-rex 3074	. . . . 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 799	. . 3 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) ↔ 𝑦 = 𝐴))
19		vex 3449	. . . 4 ⊢ 𝑦 ∈ V
20	19	elqs 8708	. . 3 ⊢ (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴))
21		velsn 4602	. . 3 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
22	18, 20, 21	3bitr4g 313	. 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ 𝑦 ∈ {𝐴}))
23	22	eqrdv 2734	1 ⊢ (𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3073  ∅c0 4282  {csn 4586   × cxp 5631  [cec 8646   / cqs 8647

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-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

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-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-cnv 5641  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ec 8650  df-qs 8654

This theorem is referenced by:  qustriv  32152