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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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