Theorem qsxpid 30927

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 487	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = [𝑥](𝐴 × 𝐴))
2		ecxpid 30925	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → [𝑥](𝐴 × 𝐴) = 𝐴)
3	2	adantr 483	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → [𝑥](𝐴 × 𝐴) = 𝐴)
4	1, 3	eqtrd 2856	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
5	4	rexlimiva 3281	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) → 𝑦 = 𝐴)
6	5	adantl 484	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
7		n0 4310	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
8	7	biimpi 218	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
9		simpl 485	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 = 𝐴)
10	2	adantl 484	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑥](𝐴 × 𝐴) = 𝐴)
11	9, 10	eqtr4d 2859	. . . . . . . . 9 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 = [𝑥](𝐴 × 𝐴))
12	11	ex 415	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑦 = [𝑥](𝐴 × 𝐴)))
13	12	ancld 553	. . . . . . 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 509	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝑦 = 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)))
16		df-rex 3144	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = [𝑥](𝐴 × 𝐴)))
17	15, 16	sylibr 236	. . . 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 3497	. . . 4 ⊢ 𝑦 ∈ V
20	19	elqs 8349	. . 3 ⊢ (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝐴 × 𝐴))
21		velsn 4583	. . 3 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
22	18, 20, 21	3bitr4g 316	. 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ 𝑦 ∈ {𝐴}))
23	22	eqrdv 2819	1 ⊢ (𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  ∃wrex 3139  ∅c0 4291  {csn 4567   × cxp 5553  [cec 8287   / cqs 8288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ec 8291  df-qs 8295

This theorem is referenced by:  qustriv  30929