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

Theorem pmtrprfval 18594
 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 5309 . . 3 {1, 2} ∈ V
2 eqid 2820 . . . 4 (pmTrsp‘{1, 2}) = (pmTrsp‘{1, 2})
32pmtrfval 18557 . . 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 10615 . . . . 5 1 ∈ V
6 2nn0 11893 . . . . 5 2 ∈ ℕ0
7 1ne2 11824 . . . . 5 1 ≠ 2
8 pr2pwpr 13822 . . . . 5 ((1 ∈ V ∧ 2 ∈ ℕ0 ∧ 1 ≠ 2) → {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} = {{1, 2}})
95, 6, 7, 8mp3an 1457 . . . 4 {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} = {{1, 2}}
109mpteq1i 5132 . . 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 4560 . . . . . 6 (𝑝 ∈ {{1, 2}} → 𝑝 = {1, 2})
12 eleq2 2899 . . . . . . . . 9 (𝑝 = {1, 2} → (𝑧𝑝𝑧 ∈ {1, 2}))
1312biimpar 480 . . . . . . . 8 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → 𝑧𝑝)
1413iftrued 4451 . . . . . . 7 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = (𝑝 ∖ {𝑧}))
15 elpri 4565 . . . . . . . . 9 (𝑧 ∈ {1, 2} → (𝑧 = 1 ∨ 𝑧 = 2))
16 2ex 11693 . . . . . . . . . . . . 13 2 ∈ V
1716unisn 4834 . . . . . . . . . . . 12 {2} = 2
18 simpr 487 . . . . . . . . . . . . . . 15 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
19 sneq 4553 . . . . . . . . . . . . . . . 16 (𝑧 = 1 → {𝑧} = {1})
2019adantr 483 . . . . . . . . . . . . . . 15 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → {𝑧} = {1})
2118, 20difeq12d 4079 . . . . . . . . . . . . . 14 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {1}))
22 difprsn1 4709 . . . . . . . . . . . . . . 15 (1 ≠ 2 → ({1, 2} ∖ {1}) = {2})
237, 22ax-mp 5 . . . . . . . . . . . . . 14 ({1, 2} ∖ {1}) = {2}
2421, 23syl6eq 2871 . . . . . . . . . . . . 13 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
2524unieqd 4828 . . . . . . . . . . . 12 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
26 iftrue 4449 . . . . . . . . . . . . 13 (𝑧 = 1 → if(𝑧 = 1, 2, 1) = 2)
2726adantr 483 . . . . . . . . . . . 12 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 2)
2817, 25, 273eqtr4a 2881 . . . . . . . . . . 11 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
2928ex 415 . . . . . . . . . 10 (𝑧 = 1 → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
305unisn 4834 . . . . . . . . . . . 12 {1} = 1
31 simpr 487 . . . . . . . . . . . . . . 15 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
32 sneq 4553 . . . . . . . . . . . . . . . 16 (𝑧 = 2 → {𝑧} = {2})
3332adantr 483 . . . . . . . . . . . . . . 15 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → {𝑧} = {2})
3431, 33difeq12d 4079 . . . . . . . . . . . . . 14 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {2}))
35 difprsn2 4710 . . . . . . . . . . . . . . 15 (1 ≠ 2 → ({1, 2} ∖ {2}) = {1})
367, 35ax-mp 5 . . . . . . . . . . . . . 14 ({1, 2} ∖ {2}) = {1}
3734, 36syl6eq 2871 . . . . . . . . . . . . 13 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
3837unieqd 4828 . . . . . . . . . . . 12 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
397nesymi 3063 . . . . . . . . . . . . . . 15 ¬ 2 = 1
40 eqeq1 2824 . . . . . . . . . . . . . . 15 (𝑧 = 2 → (𝑧 = 1 ↔ 2 = 1))
4139, 40mtbiri 329 . . . . . . . . . . . . . 14 (𝑧 = 2 → ¬ 𝑧 = 1)
4241iffalsed 4454 . . . . . . . . . . . . 13 (𝑧 = 2 → if(𝑧 = 1, 2, 1) = 1)
4342adantr 483 . . . . . . . . . . . 12 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 1)
4430, 38, 433eqtr4a 2881 . . . . . . . . . . 11 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
4544ex 415 . . . . . . . . . 10 (𝑧 = 2 → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
4629, 45jaoi 853 . . . . . . . . 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 410 . . . . . . 7 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
4914, 48eqtrd 2855 . . . . . 6 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
5011, 49sylan 582 . . . . 5 ((𝑝 ∈ {{1, 2}} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
5150mpteq2dva 5137 . . . 4 (𝑝 ∈ {{1, 2}} → (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
5251mpteq2ia 5133 . . 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 2843 . 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 2843 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 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ≠ wne 3006  {crab 3129  Vcvv 3473   ∖ cdif 3910  ifcif 4443  𝒫 cpw 4515  {csn 4543  {cpr 4545  ∪ cuni 4814   class class class wbr 5042   ↦ cmpt 5122  ‘cfv 6331  2oc2o 8074   ≈ cen 8484  1c1 10516  2c2 11671  ℕ0cn0 11876  pmTrspcpmtr 18548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-2o 8081  df-oadd 8084  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-dju 9308  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-nn 11617  df-2 11679  df-n0 11877  df-z 11961  df-uz 12223  df-fz 12877  df-hash 13676  df-pmtr 18549 This theorem is referenced by:  pmtrprfvalrn  18595
 Copyright terms: Public domain W3C validator