MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pmtrprfval Structured version   Visualization version   GIF version

Theorem pmtrprfval 18544
Description: The transpositions on a pair. (Contributed by AV, 9-Dec-2018.)
Assertion
Ref Expression
pmtrprfval (pmTrsp‘{1, 2}) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
Distinct variable group:   𝑧,𝑝

Proof of Theorem pmtrprfval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 prex 5323 . . 3 {1, 2} ∈ V
2 eqid 2818 . . . 4 (pmTrsp‘{1, 2}) = (pmTrsp‘{1, 2})
32pmtrfval 18507 . . 3 ({1, 2} ∈ V → (pmTrsp‘{1, 2}) = (𝑝 ∈ {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
41, 3ax-mp 5 . 2 (pmTrsp‘{1, 2}) = (𝑝 ∈ {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
5 1ex 10625 . . . . 5 1 ∈ V
6 2nn0 11902 . . . . 5 2 ∈ ℕ0
7 1ne2 11833 . . . . 5 1 ≠ 2
8 pr2pwpr 13825 . . . . 5 ((1 ∈ V ∧ 2 ∈ ℕ0 ∧ 1 ≠ 2) → {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} = {{1, 2}})
95, 6, 7, 8mp3an 1452 . . . 4 {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} = {{1, 2}}
109mpteq1i 5147 . . 3 (𝑝 ∈ {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
11 elsni 4574 . . . . . 6 (𝑝 ∈ {{1, 2}} → 𝑝 = {1, 2})
12 eleq2 2898 . . . . . . . . 9 (𝑝 = {1, 2} → (𝑧𝑝𝑧 ∈ {1, 2}))
1312biimpar 478 . . . . . . . 8 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → 𝑧𝑝)
1413iftrued 4471 . . . . . . 7 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = (𝑝 ∖ {𝑧}))
15 elpri 4579 . . . . . . . . 9 (𝑧 ∈ {1, 2} → (𝑧 = 1 ∨ 𝑧 = 2))
16 2ex 11702 . . . . . . . . . . . . 13 2 ∈ V
1716unisn 4846 . . . . . . . . . . . 12 {2} = 2
18 simpr 485 . . . . . . . . . . . . . . 15 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
19 sneq 4567 . . . . . . . . . . . . . . . 16 (𝑧 = 1 → {𝑧} = {1})
2019adantr 481 . . . . . . . . . . . . . . 15 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → {𝑧} = {1})
2118, 20difeq12d 4097 . . . . . . . . . . . . . 14 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {1}))
22 difprsn1 4725 . . . . . . . . . . . . . . 15 (1 ≠ 2 → ({1, 2} ∖ {1}) = {2})
237, 22ax-mp 5 . . . . . . . . . . . . . 14 ({1, 2} ∖ {1}) = {2}
2421, 23syl6eq 2869 . . . . . . . . . . . . 13 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
2524unieqd 4840 . . . . . . . . . . . 12 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
26 iftrue 4469 . . . . . . . . . . . . 13 (𝑧 = 1 → if(𝑧 = 1, 2, 1) = 2)
2726adantr 481 . . . . . . . . . . . 12 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 2)
2817, 25, 273eqtr4a 2879 . . . . . . . . . . 11 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
2928ex 413 . . . . . . . . . 10 (𝑧 = 1 → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
305unisn 4846 . . . . . . . . . . . 12 {1} = 1
31 simpr 485 . . . . . . . . . . . . . . 15 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
32 sneq 4567 . . . . . . . . . . . . . . . 16 (𝑧 = 2 → {𝑧} = {2})
3332adantr 481 . . . . . . . . . . . . . . 15 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → {𝑧} = {2})
3431, 33difeq12d 4097 . . . . . . . . . . . . . 14 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {2}))
35 difprsn2 4726 . . . . . . . . . . . . . . 15 (1 ≠ 2 → ({1, 2} ∖ {2}) = {1})
367, 35ax-mp 5 . . . . . . . . . . . . . 14 ({1, 2} ∖ {2}) = {1}
3734, 36syl6eq 2869 . . . . . . . . . . . . 13 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
3837unieqd 4840 . . . . . . . . . . . 12 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
397nesymi 3070 . . . . . . . . . . . . . . 15 ¬ 2 = 1
40 eqeq1 2822 . . . . . . . . . . . . . . 15 (𝑧 = 2 → (𝑧 = 1 ↔ 2 = 1))
4139, 40mtbiri 328 . . . . . . . . . . . . . 14 (𝑧 = 2 → ¬ 𝑧 = 1)
4241iffalsed 4474 . . . . . . . . . . . . 13 (𝑧 = 2 → if(𝑧 = 1, 2, 1) = 1)
4342adantr 481 . . . . . . . . . . . 12 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 1)
4430, 38, 433eqtr4a 2879 . . . . . . . . . . 11 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
4544ex 413 . . . . . . . . . 10 (𝑧 = 2 → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
4629, 45jaoi 851 . . . . . . . . 9 ((𝑧 = 1 ∨ 𝑧 = 2) → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
4715, 46syl 17 . . . . . . . 8 (𝑧 ∈ {1, 2} → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
4847impcom 408 . . . . . . 7 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
4914, 48eqtrd 2853 . . . . . 6 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
5011, 49sylan 580 . . . . 5 ((𝑝 ∈ {{1, 2}} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
5150mpteq2dva 5152 . . . 4 (𝑝 ∈ {{1, 2}} → (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
5251mpteq2ia 5148 . . 3 (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
5310, 52eqtri 2841 . 2 (𝑝 ∈ {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
544, 53eqtri 2841 1 (pmTrsp‘{1, 2}) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3013  {crab 3139  Vcvv 3492  cdif 3930  ifcif 4463  𝒫 cpw 4535  {csn 4557  {cpr 4559   cuni 4830   class class class wbr 5057  cmpt 5137  cfv 6348  2oc2o 8085  cen 8494  1c1 10526  2c2 11680  0cn0 11885  pmTrspcpmtr 18498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-hash 13679  df-pmtr 18499
This theorem is referenced by:  pmtrprfvalrn  18545
  Copyright terms: Public domain W3C validator