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Theorem qsxpid 32474
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.
StepHypRef Expression
1 simpr 486 . . . . . . 7 ((𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = [𝑥](𝐴 × 𝐴))
2 ecxpid 32472 . . . . . . . 8 (𝑥𝐴 → [𝑥](𝐴 × 𝐴) = 𝐴)
32adantr 482 . . . . . . 7 ((𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)) → [𝑥](𝐴 × 𝐴) = 𝐴)
41, 3eqtrd 2773 . . . . . 6 ((𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
54rexlimiva 3148 . . . . 5 (∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴) → 𝑦 = 𝐴)
65adantl 483 . . . 4 ((𝐴 ≠ ∅ ∧ ∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
7 n0 4347 . . . . . . 7 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
87biimpi 215 . . . . . 6 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
9 simpl 484 . . . . . . . . . 10 ((𝑦 = 𝐴𝑥𝐴) → 𝑦 = 𝐴)
102adantl 483 . . . . . . . . . 10 ((𝑦 = 𝐴𝑥𝐴) → [𝑥](𝐴 × 𝐴) = 𝐴)
119, 10eqtr4d 2776 . . . . . . . . 9 ((𝑦 = 𝐴𝑥𝐴) → 𝑦 = [𝑥](𝐴 × 𝐴))
1211ex 414 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)))
1312ancld 552 . . . . . . 7 (𝑦 = 𝐴 → (𝑥𝐴 → (𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴))))
1413eximdv 1921 . . . . . 6 (𝑦 = 𝐴 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴))))
158, 14mpan9 508 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝑦 = 𝐴) → ∃𝑥(𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)))
16 df-rex 3072 . . . . 5 (∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)))
1715, 16sylibr 233 . . . 4 ((𝐴 ≠ ∅ ∧ 𝑦 = 𝐴) → ∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴))
186, 17impbida 800 . . 3 (𝐴 ≠ ∅ → (∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴) ↔ 𝑦 = 𝐴))
19 vex 3479 . . . 4 𝑦 ∈ V
2019elqs 8763 . . 3 (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ ∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴))
21 velsn 4645 . . 3 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
2218, 20, 213bitr4g 314 . 2 (𝐴 ≠ ∅ → (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ 𝑦 ∈ {𝐴}))
2322eqrdv 2731 1 (𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2941  wrex 3071  c0 4323  {csn 4629   × cxp 5675  [cec 8701   / cqs 8702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ec 8705  df-qs 8709
This theorem is referenced by:  qustriv  32476
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