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Theorem qsalrel 42260
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 42257 . 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 3200 . . . . . . . 8 (𝜑 → ∀𝑥𝐴𝑦𝐴 𝑥 𝑦)
65adantr 480 . . . . . . 7 ((𝜑𝑎𝐴) → ∀𝑥𝐴𝑦𝐴 𝑥 𝑦)
7 simpr 484 . . . . . . . . 9 ((𝜑𝑎𝐴) → 𝑎𝐴)
8 breq1 5151 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑥 𝑦𝑎 𝑦))
98ralbidv 3176 . . . . . . . . . 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 5152 . . . . . . . . . . 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 8797 . . . . 5 ((𝜑𝑎𝐴) → [𝑎] = [𝑁] )
2019mpteq2dva 5248 . . . 4 (𝜑 → (𝑎𝐴 ↦ [𝑎] ) = (𝑎𝐴 ↦ [𝑁] ))
2120rneqd 5952 . . 3 (𝜑 → ran (𝑎𝐴 ↦ [𝑎] ) = ran (𝑎𝐴 ↦ [𝑁] ))
22 eqid 2735 . . . 4 (𝑎𝐴 ↦ [𝑁] ) = (𝑎𝐴 ↦ [𝑁] )
2312ne0d 4348 . . . 4 (𝜑𝐴 ≠ ∅)
2422, 23rnmptc 7227 . . 3 (𝜑 → ran (𝑎𝐴 ↦ [𝑁] ) = {[𝑁] })
252ecss 8792 . . . . 5 (𝜑 → [𝑁] 𝐴)
263, 18ersym 8756 . . . . . 6 ((𝜑𝑎𝐴) → 𝑁 𝑎)
2712adantr 480 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑁𝐴)
28 elecg 8788 . . . . . . 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 4017 . . . 4 (𝜑 → [𝑁] = 𝐴)
3231sneqd 4643 . . 3 (𝜑 → {[𝑁] } = {𝐴})
3321, 24, 323eqtrd 2779 . 2 (𝜑 → ran (𝑎𝐴 ↦ [𝑎] ) = {𝐴})
341, 33eqtrid 2787 1 (𝜑 → (𝐴 / ) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {csn 4631   class class class wbr 5148  cmpt 5231  ran crn 5690   Er wer 8741  [cec 8742   / cqs 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-er 8744  df-ec 8746  df-qs 8750
This theorem is referenced by:  0prjspn  42615
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