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Theorem pmtrsn 17855
Description: The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for 𝐴 ∉ V, i.e. for the empty set {𝐴} = ∅ resulting in (pmTrsp‘∅) = ∅. (Contributed by AV, 6-Aug-2019.)
Assertion
Ref Expression
pmtrsn (pmTrsp‘{𝐴}) = ∅

Proof of Theorem pmtrsn
Dummy variables 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4874 . . 3 {𝐴} ∈ V
2 eqid 2626 . . . 4 (pmTrsp‘{𝐴}) = (pmTrsp‘{𝐴})
32pmtrfval 17786 . . 3 ({𝐴} ∈ V → (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
41, 3ax-mp 5 . 2 (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
5 eqid 2626 . . . . 5 (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
65dmmpt 5592 . . . 4 dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
7 2on0 7515 . . . . . . . . 9 2𝑜 ≠ ∅
8 ensymb 7949 . . . . . . . . . 10 (∅ ≈ 2𝑜 ↔ 2𝑜 ≈ ∅)
9 en0 7964 . . . . . . . . . 10 (2𝑜 ≈ ∅ ↔ 2𝑜 = ∅)
108, 9bitri 264 . . . . . . . . 9 (∅ ≈ 2𝑜 ↔ 2𝑜 = ∅)
117, 10nemtbir 2891 . . . . . . . 8 ¬ ∅ ≈ 2𝑜
12 snnen2o 8094 . . . . . . . 8 ¬ {𝐴} ≈ 2𝑜
13 0ex 4755 . . . . . . . . 9 ∅ ∈ V
14 breq1 4621 . . . . . . . . . 10 (𝑦 = ∅ → (𝑦 ≈ 2𝑜 ↔ ∅ ≈ 2𝑜))
1514notbid 308 . . . . . . . . 9 (𝑦 = ∅ → (¬ 𝑦 ≈ 2𝑜 ↔ ¬ ∅ ≈ 2𝑜))
16 breq1 4621 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝑦 ≈ 2𝑜 ↔ {𝐴} ≈ 2𝑜))
1716notbid 308 . . . . . . . . 9 (𝑦 = {𝐴} → (¬ 𝑦 ≈ 2𝑜 ↔ ¬ {𝐴} ≈ 2𝑜))
1813, 1, 15, 17ralpr 4214 . . . . . . . 8 (∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2𝑜 ↔ (¬ ∅ ≈ 2𝑜 ∧ ¬ {𝐴} ≈ 2𝑜))
1911, 12, 18mpbir2an 954 . . . . . . 7 𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2𝑜
20 pwsn 4401 . . . . . . . 8 𝒫 {𝐴} = {∅, {𝐴}}
2120raleqi 3136 . . . . . . 7 (∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2𝑜 ↔ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2𝑜)
2219, 21mpbir 221 . . . . . 6 𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2𝑜
23 rabeq0 3936 . . . . . 6 ({𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} = ∅ ↔ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2𝑜)
2422, 23mpbir 221 . . . . 5 {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} = ∅
25 rabeq 3184 . . . . 5 ({𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} = ∅ → {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V})
2624, 25ax-mp 5 . . . 4 {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
27 rab0 3934 . . . 4 {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = ∅
286, 26, 273eqtri 2652 . . 3 dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅
29 funmpt 5886 . . . . 5 Fun (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
30 funrel 5867 . . . . 5 (Fun (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
3129, 30ax-mp 5 . . . 4 Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
32 reldm0 5307 . . . 4 (Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅))
3331, 32ax-mp 5 . . 3 ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅)
3428, 33mpbir 221 . 2 (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅
354, 34eqtri 2648 1 (pmTrsp‘{𝐴}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1480  wcel 1992  wral 2912  {crab 2916  Vcvv 3191  cdif 3557  c0 3896  ifcif 4063  𝒫 cpw 4135  {csn 4153  {cpr 4155   cuni 4407   class class class wbr 4618  cmpt 4678  dom cdm 5079  Rel wrel 5084  Fun wfun 5844  cfv 5850  2𝑜c2o 7500  cen 7897  pmTrspcpmtr 17777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-1o 7506  df-2o 7507  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pmtr 17778
This theorem is referenced by:  psgnsn  17856
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