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Theorem pmtrprfval 18115
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 5038 . . 3 {1, 2} ∈ V
2 eqid 2771 . . . 4 (pmTrsp‘{1, 2}) = (pmTrsp‘{1, 2})
32pmtrfval 18078 . . 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 10238 . . . . 5 1 ∈ V
6 2nn0 11512 . . . . 5 2 ∈ ℕ0
7 1ne2 11443 . . . . 5 1 ≠ 2
8 pr2pwpr 13464 . . . . 5 ((1 ∈ V ∧ 2 ∈ ℕ0 ∧ 1 ≠ 2) → {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2𝑜} = {{1, 2}})
95, 6, 7, 8mp3an 1572 . . . 4 {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2𝑜} = {{1, 2}}
10 eqid 2771 . . . 4 (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))
119, 10mpteq12i 4877 . . 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)))
12 elsni 4334 . . . . . 6 (𝑝 ∈ {{1, 2}} → 𝑝 = {1, 2})
13 eleq2 2839 . . . . . . . . 9 (𝑝 = {1, 2} → (𝑧𝑝𝑧 ∈ {1, 2}))
1413biimpar 463 . . . . . . . 8 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → 𝑧𝑝)
1514iftrued 4234 . . . . . . 7 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = (𝑝 ∖ {𝑧}))
16 elpri 4338 . . . . . . . . 9 (𝑧 ∈ {1, 2} → (𝑧 = 1 ∨ 𝑧 = 2))
17 2ex 11295 . . . . . . . . . . . . 13 2 ∈ V
1817unisn 4590 . . . . . . . . . . . 12 {2} = 2
19 simpr 471 . . . . . . . . . . . . . . 15 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
20 sneq 4327 . . . . . . . . . . . . . . . 16 (𝑧 = 1 → {𝑧} = {1})
2120adantr 466 . . . . . . . . . . . . . . 15 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → {𝑧} = {1})
2219, 21difeq12d 3881 . . . . . . . . . . . . . 14 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {1}))
23 difprsn1 4467 . . . . . . . . . . . . . . 15 (1 ≠ 2 → ({1, 2} ∖ {1}) = {2})
247, 23ax-mp 5 . . . . . . . . . . . . . 14 ({1, 2} ∖ {1}) = {2}
2522, 24syl6eq 2821 . . . . . . . . . . . . 13 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
2625unieqd 4585 . . . . . . . . . . . 12 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
27 iftrue 4232 . . . . . . . . . . . . 13 (𝑧 = 1 → if(𝑧 = 1, 2, 1) = 2)
2827adantr 466 . . . . . . . . . . . 12 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 2)
2918, 26, 283eqtr4a 2831 . . . . . . . . . . 11 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
3029ex 397 . . . . . . . . . 10 (𝑧 = 1 → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
315unisn 4590 . . . . . . . . . . . 12 {1} = 1
32 simpr 471 . . . . . . . . . . . . . . 15 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
33 sneq 4327 . . . . . . . . . . . . . . . 16 (𝑧 = 2 → {𝑧} = {2})
3433adantr 466 . . . . . . . . . . . . . . 15 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → {𝑧} = {2})
3532, 34difeq12d 3881 . . . . . . . . . . . . . 14 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {2}))
36 difprsn2 4468 . . . . . . . . . . . . . . 15 (1 ≠ 2 → ({1, 2} ∖ {2}) = {1})
377, 36ax-mp 5 . . . . . . . . . . . . . 14 ({1, 2} ∖ {2}) = {1}
3835, 37syl6eq 2821 . . . . . . . . . . . . 13 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
3938unieqd 4585 . . . . . . . . . . . 12 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
407nesymi 3000 . . . . . . . . . . . . . . 15 ¬ 2 = 1
41 eqeq1 2775 . . . . . . . . . . . . . . 15 (𝑧 = 2 → (𝑧 = 1 ↔ 2 = 1))
4240, 41mtbiri 316 . . . . . . . . . . . . . 14 (𝑧 = 2 → ¬ 𝑧 = 1)
4342iffalsed 4237 . . . . . . . . . . . . 13 (𝑧 = 2 → if(𝑧 = 1, 2, 1) = 1)
4443adantr 466 . . . . . . . . . . . 12 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 1)
4531, 39, 443eqtr4a 2831 . . . . . . . . . . 11 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
4645ex 397 . . . . . . . . . 10 (𝑧 = 2 → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
4730, 46jaoi 838 . . . . . . . . 9 ((𝑧 = 1 ∨ 𝑧 = 2) → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
4816, 47syl 17 . . . . . . . 8 (𝑧 ∈ {1, 2} → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
4948impcom 394 . . . . . . 7 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
5015, 49eqtrd 2805 . . . . . 6 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
5112, 50sylan 563 . . . . 5 ((𝑝 ∈ {{1, 2}} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
5251mpteq2dva 4879 . . . 4 (𝑝 ∈ {{1, 2}} → (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
5352mpteq2ia 4875 . . 3 (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
5411, 53eqtri 2793 . 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)))
554, 54eqtri 2793 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 382  wo 828   = wceq 1631  wcel 2145  wne 2943  {crab 3065  Vcvv 3351  cdif 3721  ifcif 4226  𝒫 cpw 4298  {csn 4317  {cpr 4319   cuni 4575   class class class wbr 4787  cmpt 4864  cfv 6032  2𝑜c2o 7708  cen 8107  1c1 10140  2c2 11273  0cn0 11495  pmTrspcpmtr 18069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-cnex 10195  ax-resscn 10196  ax-1cn 10197  ax-icn 10198  ax-addcl 10199  ax-addrcl 10200  ax-mulcl 10201  ax-mulrcl 10202  ax-mulcom 10203  ax-addass 10204  ax-mulass 10205  ax-distr 10206  ax-i2m1 10207  ax-1ne0 10208  ax-1rid 10209  ax-rnegex 10210  ax-rrecex 10211  ax-cnre 10212  ax-pre-lttri 10213  ax-pre-lttrn 10214  ax-pre-ltadd 10215  ax-pre-mulgt0 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-2o 7715  df-oadd 7718  df-er 7897  df-en 8111  df-dom 8112  df-sdom 8113  df-fin 8114  df-card 8966  df-cda 9193  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282  df-le 10283  df-sub 10471  df-neg 10472  df-nn 11224  df-2 11282  df-n0 11496  df-z 11581  df-uz 11890  df-fz 12535  df-hash 13323  df-pmtr 18070
This theorem is referenced by:  pmtrprfvalrn  18116
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