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Theorem pmtrsn 18131
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 5049 . . 3 {𝐴} ∈ V
2 eqid 2752 . . . 4 (pmTrsp‘{𝐴}) = (pmTrsp‘{𝐴})
32pmtrfval 18062 . . 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 2752 . . . . 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 5783 . . . 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 7730 . . . . . . . . 9 2𝑜 ≠ ∅
8 ensymb 8161 . . . . . . . . . 10 (∅ ≈ 2𝑜 ↔ 2𝑜 ≈ ∅)
9 en0 8176 . . . . . . . . . 10 (2𝑜 ≈ ∅ ↔ 2𝑜 = ∅)
108, 9bitri 264 . . . . . . . . 9 (∅ ≈ 2𝑜 ↔ 2𝑜 = ∅)
117, 10nemtbir 3019 . . . . . . . 8 ¬ ∅ ≈ 2𝑜
12 snnen2o 8306 . . . . . . . 8 ¬ {𝐴} ≈ 2𝑜
13 0ex 4934 . . . . . . . . 9 ∅ ∈ V
14 breq1 4799 . . . . . . . . . 10 (𝑦 = ∅ → (𝑦 ≈ 2𝑜 ↔ ∅ ≈ 2𝑜))
1514notbid 307 . . . . . . . . 9 (𝑦 = ∅ → (¬ 𝑦 ≈ 2𝑜 ↔ ¬ ∅ ≈ 2𝑜))
16 breq1 4799 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝑦 ≈ 2𝑜 ↔ {𝐴} ≈ 2𝑜))
1716notbid 307 . . . . . . . . 9 (𝑦 = {𝐴} → (¬ 𝑦 ≈ 2𝑜 ↔ ¬ {𝐴} ≈ 2𝑜))
1813, 1, 15, 17ralpr 4374 . . . . . . . 8 (∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2𝑜 ↔ (¬ ∅ ≈ 2𝑜 ∧ ¬ {𝐴} ≈ 2𝑜))
1911, 12, 18mpbir2an 993 . . . . . . 7 𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2𝑜
20 pwsn 4572 . . . . . . . 8 𝒫 {𝐴} = {∅, {𝐴}}
2120raleqi 3273 . . . . . . 7 (∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2𝑜 ↔ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2𝑜)
2219, 21mpbir 221 . . . . . 6 𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2𝑜
23 rabeq0 4092 . . . . . 6 ({𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} = ∅ ↔ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2𝑜)
2422, 23mpbir 221 . . . . 5 {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} = ∅
25 rabeq 3324 . . . . 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 4090 . . . 4 {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = ∅
286, 26, 273eqtri 2778 . . 3 dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = ∅
29 funmpt 6079 . . . . 5 Fun (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
30 funrel 6058 . . . . 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 5490 . . . 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 2774 1 (pmTrsp‘{𝐴}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1624  wcel 2131  wral 3042  {crab 3046  Vcvv 3332  cdif 3704  c0 4050  ifcif 4222  𝒫 cpw 4294  {csn 4313  {cpr 4315   cuni 4580   class class class wbr 4796  cmpt 4873  dom cdm 5258  Rel wrel 5263  Fun wfun 6035  cfv 6041  2𝑜c2o 7715  cen 8110  pmTrspcpmtr 18053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-1o 7721  df-2o 7722  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-pmtr 18054
This theorem is referenced by:  psgnsn  18132
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