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Theorem qsalrel 40215
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 40212 . 2 (𝐴 / ) = ran (𝑎𝐴 ↦ [𝑎] )
2 qsalrel.2 . . . . . . 7 (𝜑 Er 𝐴)
32adantr 481 . . . . . 6 ((𝜑𝑎𝐴) → Er 𝐴)
4 qsalrel.1 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 𝑦)
54ralrimivva 3123 . . . . . . . 8 (𝜑 → ∀𝑥𝐴𝑦𝐴 𝑥 𝑦)
65adantr 481 . . . . . . 7 ((𝜑𝑎𝐴) → ∀𝑥𝐴𝑦𝐴 𝑥 𝑦)
7 simpr 485 . . . . . . . . 9 ((𝜑𝑎𝐴) → 𝑎𝐴)
8 breq1 5077 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑥 𝑦𝑎 𝑦))
98ralbidv 3112 . . . . . . . . . 10 (𝑥 = 𝑎 → (∀𝑦𝐴 𝑥 𝑦 ↔ ∀𝑦𝐴 𝑎 𝑦))
109adantl 482 . . . . . . . . 9 (((𝜑𝑎𝐴) ∧ 𝑥 = 𝑎) → (∀𝑦𝐴 𝑥 𝑦 ↔ ∀𝑦𝐴 𝑎 𝑦))
117, 10rspcdv 3553 . . . . . . . 8 ((𝜑𝑎𝐴) → (∀𝑥𝐴𝑦𝐴 𝑥 𝑦 → ∀𝑦𝐴 𝑎 𝑦))
12 qsalrel.3 . . . . . . . . . 10 (𝜑𝑁𝐴)
13 breq2 5078 . . . . . . . . . . 11 (𝑦 = 𝑁 → (𝑎 𝑦𝑎 𝑁))
1413adantl 482 . . . . . . . . . 10 ((𝜑𝑦 = 𝑁) → (𝑎 𝑦𝑎 𝑁))
1512, 14rspcdv 3553 . . . . . . . . 9 (𝜑 → (∀𝑦𝐴 𝑎 𝑦𝑎 𝑁))
1615adantr 481 . . . . . . . 8 ((𝜑𝑎𝐴) → (∀𝑦𝐴 𝑎 𝑦𝑎 𝑁))
1711, 16syld 47 . . . . . . 7 ((𝜑𝑎𝐴) → (∀𝑥𝐴𝑦𝐴 𝑥 𝑦𝑎 𝑁))
186, 17mpd 15 . . . . . 6 ((𝜑𝑎𝐴) → 𝑎 𝑁)
193, 18erthi 8549 . . . . 5 ((𝜑𝑎𝐴) → [𝑎] = [𝑁] )
2019mpteq2dva 5174 . . . 4 (𝜑 → (𝑎𝐴 ↦ [𝑎] ) = (𝑎𝐴 ↦ [𝑁] ))
2120rneqd 5847 . . 3 (𝜑 → ran (𝑎𝐴 ↦ [𝑎] ) = ran (𝑎𝐴 ↦ [𝑁] ))
22 eqid 2738 . . . 4 (𝑎𝐴 ↦ [𝑁] ) = (𝑎𝐴 ↦ [𝑁] )
2312ne0d 4269 . . . 4 (𝜑𝐴 ≠ ∅)
2422, 23rnmptc 7082 . . 3 (𝜑 → ran (𝑎𝐴 ↦ [𝑁] ) = {[𝑁] })
252ecss 8544 . . . . 5 (𝜑 → [𝑁] 𝐴)
263, 18ersym 8510 . . . . . 6 ((𝜑𝑎𝐴) → 𝑁 𝑎)
2712adantr 481 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑁𝐴)
28 elecg 8541 . . . . . . 7 ((𝑎𝐴𝑁𝐴) → (𝑎 ∈ [𝑁] 𝑁 𝑎))
297, 27, 28syl2anc 584 . . . . . 6 ((𝜑𝑎𝐴) → (𝑎 ∈ [𝑁] 𝑁 𝑎))
3026, 29mpbird 256 . . . . 5 ((𝜑𝑎𝐴) → 𝑎 ∈ [𝑁] )
3125, 30eqelssd 3942 . . . 4 (𝜑 → [𝑁] = 𝐴)
3231sneqd 4573 . . 3 (𝜑 → {[𝑁] } = {𝐴})
3321, 24, 323eqtrd 2782 . 2 (𝜑 → ran (𝑎𝐴 ↦ [𝑎] ) = {𝐴})
341, 33eqtrid 2790 1 (𝜑 → (𝐴 / ) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  {csn 4561   class class class wbr 5074  cmpt 5157  ran crn 5590   Er wer 8495  [cec 8496   / cqs 8497
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-mpt 5158  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-er 8498  df-ec 8500  df-qs 8504
This theorem is referenced by:  0prjspn  40465
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