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Theorem qsalrel 42281
Description: The quotient set is equal to the singleton of 𝐴 when all elements are related and 𝐴 is nonempty. (Contributed by SN, 8-Jun-2023.)
Hypotheses
Ref Expression
qsalrel.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 𝑦)
qsalrel.2 (𝜑 Er 𝐴)
qsalrel.3 (𝜑𝑁𝐴)
Assertion
Ref Expression
qsalrel (𝜑 → (𝐴 / ) = {𝐴})
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑥, ,𝑦   𝑦,𝑁
Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem qsalrel
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 dfqs2 42278 . 2 (𝐴 / ) = ran (𝑎𝐴 ↦ [𝑎] )
2 qsalrel.2 . . . . . . 7 (𝜑 Er 𝐴)
32adantr 480 . . . . . 6 ((𝜑𝑎𝐴) → Er 𝐴)
4 qsalrel.1 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 𝑦)
54ralrimivva 3202 . . . . . . . 8 (𝜑 → ∀𝑥𝐴𝑦𝐴 𝑥 𝑦)
65adantr 480 . . . . . . 7 ((𝜑𝑎𝐴) → ∀𝑥𝐴𝑦𝐴 𝑥 𝑦)
7 simpr 484 . . . . . . . . 9 ((𝜑𝑎𝐴) → 𝑎𝐴)
8 breq1 5146 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑥 𝑦𝑎 𝑦))
98ralbidv 3178 . . . . . . . . . 10 (𝑥 = 𝑎 → (∀𝑦𝐴 𝑥 𝑦 ↔ ∀𝑦𝐴 𝑎 𝑦))
109adantl 481 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑥 = 𝑎) → (∀𝑦𝐴 𝑥 𝑦 ↔ ∀𝑦𝐴 𝑎 𝑦))
117, 10rspcdv 3614 . . . . . . . 8 ((𝜑𝑎𝐴) → (∀𝑥𝐴𝑦𝐴 𝑥 𝑦 → ∀𝑦𝐴 𝑎 𝑦))
12 qsalrel.3 . . . . . . . . . 10 (𝜑𝑁𝐴)
13 breq2 5147 . . . . . . . . . . 11 (𝑦 = 𝑁 → (𝑎 𝑦𝑎 𝑁))
1413adantl 481 . . . . . . . . . 10 ((𝜑𝑦 = 𝑁) → (𝑎 𝑦𝑎 𝑁))
1512, 14rspcdv 3614 . . . . . . . . 9 (𝜑 → (∀𝑦𝐴 𝑎 𝑦𝑎 𝑁))
1615adantr 480 . . . . . . . 8 ((𝜑𝑎𝐴) → (∀𝑦𝐴 𝑎 𝑦𝑎 𝑁))
1711, 16syld 47 . . . . . . 7 ((𝜑𝑎𝐴) → (∀𝑥𝐴𝑦𝐴 𝑥 𝑦𝑎 𝑁))
186, 17mpd 15 . . . . . 6 ((𝜑𝑎𝐴) → 𝑎 𝑁)
193, 18erthi 8798 . . . . 5 ((𝜑𝑎𝐴) → [𝑎] = [𝑁] )
2019mpteq2dva 5242 . . . 4 (𝜑 → (𝑎𝐴 ↦ [𝑎] ) = (𝑎𝐴 ↦ [𝑁] ))
2120rneqd 5949 . . 3 (𝜑 → ran (𝑎𝐴 ↦ [𝑎] ) = ran (𝑎𝐴 ↦ [𝑁] ))
22 eqid 2737 . . . 4 (𝑎𝐴 ↦ [𝑁] ) = (𝑎𝐴 ↦ [𝑁] )
2312ne0d 4342 . . . 4 (𝜑𝐴 ≠ ∅)
2422, 23rnmptc 7227 . . 3 (𝜑 → ran (𝑎𝐴 ↦ [𝑁] ) = {[𝑁] })
252ecss 8793 . . . . 5 (𝜑 → [𝑁] 𝐴)
263, 18ersym 8757 . . . . . 6 ((𝜑𝑎𝐴) → 𝑁 𝑎)
2712adantr 480 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑁𝐴)
28 elecg 8789 . . . . . . 7 ((𝑎𝐴𝑁𝐴) → (𝑎 ∈ [𝑁] 𝑁 𝑎))
297, 27, 28syl2anc 584 . . . . . 6 ((𝜑𝑎𝐴) → (𝑎 ∈ [𝑁] 𝑁 𝑎))
3026, 29mpbird 257 . . . . 5 ((𝜑𝑎𝐴) → 𝑎 ∈ [𝑁] )
3125, 30eqelssd 4005 . . . 4 (𝜑 → [𝑁] = 𝐴)
3231sneqd 4638 . . 3 (𝜑 → {[𝑁] } = {𝐴})
3321, 24, 323eqtrd 2781 . 2 (𝜑 → ran (𝑎𝐴 ↦ [𝑎] ) = {𝐴})
341, 33eqtrid 2789 1 (𝜑 → (𝐴 / ) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {csn 4626   class class class wbr 5143  cmpt 5225  ran crn 5686   Er wer 8742  [cec 8743   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-er 8745  df-ec 8747  df-qs 8751
This theorem is referenced by:  0prjspn  42638
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