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Theorem qsxpid 31083
 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 488 . . . . . . 7 ((𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = [𝑥](𝐴 × 𝐴))
2 ecxpid 31081 . . . . . . . 8 (𝑥𝐴 → [𝑥](𝐴 × 𝐴) = 𝐴)
32adantr 484 . . . . . . 7 ((𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)) → [𝑥](𝐴 × 𝐴) = 𝐴)
41, 3eqtrd 2793 . . . . . 6 ((𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
54rexlimiva 3205 . . . . 5 (∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴) → 𝑦 = 𝐴)
65adantl 485 . . . 4 ((𝐴 ≠ ∅ ∧ ∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴)) → 𝑦 = 𝐴)
7 n0 4247 . . . . . . 7 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
87biimpi 219 . . . . . 6 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
9 simpl 486 . . . . . . . . . 10 ((𝑦 = 𝐴𝑥𝐴) → 𝑦 = 𝐴)
102adantl 485 . . . . . . . . . 10 ((𝑦 = 𝐴𝑥𝐴) → [𝑥](𝐴 × 𝐴) = 𝐴)
119, 10eqtr4d 2796 . . . . . . . . 9 ((𝑦 = 𝐴𝑥𝐴) → 𝑦 = [𝑥](𝐴 × 𝐴))
1211ex 416 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)))
1312ancld 554 . . . . . . 7 (𝑦 = 𝐴 → (𝑥𝐴 → (𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴))))
1413eximdv 1918 . . . . . 6 (𝑦 = 𝐴 → (∃𝑥 𝑥𝐴 → ∃𝑥(𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴))))
158, 14mpan9 510 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝑦 = 𝐴) → ∃𝑥(𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)))
16 df-rex 3076 . . . . 5 (∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 = [𝑥](𝐴 × 𝐴)))
1715, 16sylibr 237 . . . 4 ((𝐴 ≠ ∅ ∧ 𝑦 = 𝐴) → ∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴))
186, 17impbida 800 . . 3 (𝐴 ≠ ∅ → (∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴) ↔ 𝑦 = 𝐴))
19 vex 3413 . . . 4 𝑦 ∈ V
2019elqs 8364 . . 3 (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ ∃𝑥𝐴 𝑦 = [𝑥](𝐴 × 𝐴))
21 velsn 4541 . . 3 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
2218, 20, 213bitr4g 317 . 2 (𝐴 ≠ ∅ → (𝑦 ∈ (𝐴 / (𝐴 × 𝐴)) ↔ 𝑦 ∈ {𝐴}))
2322eqrdv 2756 1 (𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071  ∅c0 4227  {csn 4525   × cxp 5525  [cec 8302   / cqs 8303 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-xp 5533  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-ec 8306  df-qs 8310 This theorem is referenced by:  qustriv  31085
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