Theorem qsxpid 19437

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 490	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = [𝑥](𝐴 × 𝐴))
2		ecxpid 19436	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → [𝑥](𝐴 × 𝐴) = 𝐴)
3	2	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → [𝑥](𝐴 × 𝐴) = 𝐴)
4	1, 3	eqtrd 2798	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
5	4	rexlimiva 3159	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) → 𝑦 = 𝐴)
6	5	adantl 487	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
7		n0 4303	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
8	7	biimpi 219	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
9		simpl 488	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 = 𝐴)
10	2	adantl 487	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑥](𝐴 × 𝐴) = 𝐴)
11	9, 10	eqtr4d 2801	. . . . . . . . 9 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 = [𝑥](𝐴 × 𝐴))
12	11	ex 418	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑦 = [𝑥](𝐴 × 𝐴)))
13	12	ancld 560	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴))))
14	13	eximdv 1952	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴))))
15	8, 14	mpan9 516	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝑦 = 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)))
16		df-rex 3091	. . . . 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 813	. . 3 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) ↔ 𝑦 = 𝐴))
19		vex 3458	. . . 4 ⊢ 𝑦 ∈ V
20	19	elqs 8823	. . 3 ⊢ (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴))
21		velsn 4600	. . 3 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
22	18, 20, 21	3bitr4g 317	. 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ 𝑦 ∈ {𝐴}))
23	22	eqrdv 2761	1 ⊢ (𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  ∅c0 4281  {csn 4588   × cxp 5656  [cec 8753   / cqs 8754

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757  df-qs 8761

This theorem is used by:  qustriv  19446