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Theorem pmtrprfval 17828
 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 4870 . . 3 {1, 2} ∈ V
2 eqid 2621 . . . 4 (pmTrsp‘{1, 2}) = (pmTrsp‘{1, 2})
32pmtrfval 17791 . . 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 9979 . . . . 5 1 ∈ V
6 2nn0 11253 . . . . 5 2 ∈ ℕ0
7 1ne2 11184 . . . . 5 1 ≠ 2
8 pr2pwpr 13199 . . . . 5 ((1 ∈ V ∧ 2 ∈ ℕ0 ∧ 1 ≠ 2) → {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2𝑜} = {{1, 2}})
95, 6, 7, 8mp3an 1421 . . . 4 {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2𝑜} = {{1, 2}}
10 eqid 2621 . . . 4 (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))
119, 10mpteq12i 4702 . . 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 4165 . . . . . 6 (𝑝 ∈ {{1, 2}} → 𝑝 = {1, 2})
13 eleq2 2687 . . . . . . . . 9 (𝑝 = {1, 2} → (𝑧𝑝𝑧 ∈ {1, 2}))
1413biimpar 502 . . . . . . . 8 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → 𝑧𝑝)
1514iftrued 4066 . . . . . . 7 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = (𝑝 ∖ {𝑧}))
16 elpri 4168 . . . . . . . . 9 (𝑧 ∈ {1, 2} → (𝑧 = 1 ∨ 𝑧 = 2))
17 2ex 11036 . . . . . . . . . . . . 13 2 ∈ V
1817unisn 4417 . . . . . . . . . . . 12 {2} = 2
19 simpr 477 . . . . . . . . . . . . . . 15 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
20 sneq 4158 . . . . . . . . . . . . . . . 16 (𝑧 = 1 → {𝑧} = {1})
2120adantr 481 . . . . . . . . . . . . . . 15 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → {𝑧} = {1})
2219, 21difeq12d 3707 . . . . . . . . . . . . . 14 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {1}))
23 difprsn1 4299 . . . . . . . . . . . . . . 15 (1 ≠ 2 → ({1, 2} ∖ {1}) = {2})
247, 23ax-mp 5 . . . . . . . . . . . . . 14 ({1, 2} ∖ {1}) = {2}
2522, 24syl6eq 2671 . . . . . . . . . . . . 13 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
2625unieqd 4412 . . . . . . . . . . . 12 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
27 iftrue 4064 . . . . . . . . . . . . 13 (𝑧 = 1 → if(𝑧 = 1, 2, 1) = 2)
2827adantr 481 . . . . . . . . . . . 12 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 2)
2918, 26, 283eqtr4a 2681 . . . . . . . . . . 11 ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
3029ex 450 . . . . . . . . . 10 (𝑧 = 1 → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
315unisn 4417 . . . . . . . . . . . 12 {1} = 1
32 simpr 477 . . . . . . . . . . . . . . 15 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
33 sneq 4158 . . . . . . . . . . . . . . . 16 (𝑧 = 2 → {𝑧} = {2})
3433adantr 481 . . . . . . . . . . . . . . 15 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → {𝑧} = {2})
3532, 34difeq12d 3707 . . . . . . . . . . . . . 14 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {2}))
36 difprsn2 4300 . . . . . . . . . . . . . . 15 (1 ≠ 2 → ({1, 2} ∖ {2}) = {1})
377, 36ax-mp 5 . . . . . . . . . . . . . 14 ({1, 2} ∖ {2}) = {1}
3835, 37syl6eq 2671 . . . . . . . . . . . . 13 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
3938unieqd 4412 . . . . . . . . . . . 12 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
407nesymi 2847 . . . . . . . . . . . . . . 15 ¬ 2 = 1
41 eqeq1 2625 . . . . . . . . . . . . . . 15 (𝑧 = 2 → (𝑧 = 1 ↔ 2 = 1))
4240, 41mtbiri 317 . . . . . . . . . . . . . 14 (𝑧 = 2 → ¬ 𝑧 = 1)
4342iffalsed 4069 . . . . . . . . . . . . 13 (𝑧 = 2 → if(𝑧 = 1, 2, 1) = 1)
4443adantr 481 . . . . . . . . . . . 12 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 1)
4531, 39, 443eqtr4a 2681 . . . . . . . . . . 11 ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
4645ex 450 . . . . . . . . . 10 (𝑧 = 2 → (𝑝 = {1, 2} → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
4730, 46jaoi 394 . . . . . . . . 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 446 . . . . . . 7 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
5015, 49eqtrd 2655 . . . . . 6 ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
5112, 50sylan 488 . . . . 5 ((𝑝 ∈ {{1, 2}} ∧ 𝑧 ∈ {1, 2}) → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
5251mpteq2dva 4704 . . . 4 (𝑝 ∈ {{1, 2}} → (𝑧 ∈ {1, 2} ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
5352mpteq2ia 4700 . . 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 2643 . 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 2643 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   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  {crab 2911  Vcvv 3186   ∖ cdif 3552  ifcif 4058  𝒫 cpw 4130  {csn 4148  {cpr 4150  ∪ cuni 4402   class class class wbr 4613   ↦ cmpt 4673  ‘cfv 5847  2𝑜c2o 7499   ≈ cen 7896  1c1 9881  2c2 11014  ℕ0cn0 11236  pmTrspcpmtr 17782 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-hash 13058  df-pmtr 17783 This theorem is referenced by:  pmtrprfvalrn  17829
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