Theorem pmtrprfval 18607

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.

Step	Hyp	Ref	Expression
1		prex 5298	. . 3 ⊢ {1, 2} ∈ V
2		eqid 2798	. . . 4 ⊢ (pmTrsp‘{1, 2}) = (pmTrsp‘{1, 2})
3	2	pmtrfval 18570	. . 3 ⊢ ({1, 2} ∈ V → (pmTrsp‘{1, 2}) = (𝑝 ∈ {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
4	1, 3	ax-mp 5	. 2 ⊢ (pmTrsp‘{1, 2}) = (𝑝 ∈ {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
5		1ex 10626	. . . . 5 ⊢ 1 ∈ V
6		2nn0 11902	. . . . 5 ⊢ 2 ∈ ℕ0
7		1ne2 11833	. . . . 5 ⊢ 1 ≠ 2
8		pr2pwpr 13833	. . . . 5 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ0 ∧ 1 ≠ 2) → {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} = {{1, 2}})
9	5, 6, 7, 8	mp3an 1458	. . . 4 ⊢ {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} = {{1, 2}}
10	9	mpteq1i 5120	. . 3 ⊢ (𝑝 ∈ {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
11		elsni 4542	. . . . . 6 ⊢ (𝑝 ∈ {{1, 2}} → 𝑝 = {1, 2})
12		eleq2 2878	. . . . . . . . 9 ⊢ (𝑝 = {1, 2} → (𝑧 ∈ 𝑝 ↔ 𝑧 ∈ {1, 2}))
13	12	biimpar 481	. . . . . . . 8 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → 𝑧 ∈ 𝑝)
14	13	iftrued 4433	. . . . . . 7 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = ∪ (𝑝 ∖ {𝑧}))
15		elpri 4547	. . . . . . . . 9 ⊢ (𝑧 ∈ {1, 2} → (𝑧 = 1 ∨ 𝑧 = 2))
16		2ex 11702	. . . . . . . . . . . . 13 ⊢ 2 ∈ V
17	16	unisn 4820	. . . . . . . . . . . 12 ⊢ ∪ {2} = 2
18		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
19		sneq 4535	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 1 → {𝑧} = {1})
20	19	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → {𝑧} = {1})
21	18, 20	difeq12d 4051	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {1}))
22		difprsn1 4693	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({1, 2} ∖ {1}) = {2})
23	7, 22	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({1, 2} ∖ {1}) = {2}
24	21, 23	eqtrdi 2849	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
25	24	unieqd 4814	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = ∪ {2})
26		iftrue 4431	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1 → if(𝑧 = 1, 2, 1) = 2)
27	26	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 2)
28	17, 25, 27	3eqtr4a 2859	. . . . . . . . . . 11 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
29	28	ex 416	. . . . . . . . . 10 ⊢ (𝑧 = 1 → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
30	5	unisn 4820	. . . . . . . . . . . 12 ⊢ ∪ {1} = 1
31		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
32		sneq 4535	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 2 → {𝑧} = {2})
33	32	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → {𝑧} = {2})
34	31, 33	difeq12d 4051	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {2}))
35		difprsn2 4694	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({1, 2} ∖ {2}) = {1})
36	7, 35	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({1, 2} ∖ {2}) = {1}
37	34, 36	eqtrdi 2849	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
38	37	unieqd 4814	. . . . . . . . . . . 12 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = ∪ {1})
39	7	nesymi 3044	. . . . . . . . . . . . . . 15 ⊢ ¬ 2 = 1
40		eqeq1 2802	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 2 → (𝑧 = 1 ↔ 2 = 1))
41	39, 40	mtbiri 330	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 2 → ¬ 𝑧 = 1)
42	41	iffalsed 4436	. . . . . . . . . . . . 13 ⊢ (𝑧 = 2 → if(𝑧 = 1, 2, 1) = 1)
43	42	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 1)
44	30, 38, 43	3eqtr4a 2859	. . . . . . . . . . 11 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
45	44	ex 416	. . . . . . . . . 10 ⊢ (𝑧 = 2 → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
46	29, 45	jaoi 854	. . . . . . . . 9 ⊢ ((𝑧 = 1 ∨ 𝑧 = 2) → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
47	15, 46	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ {1, 2} → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
48	47	impcom 411	. . . . . . 7 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
49	14, 48	eqtrd 2833	. . . . . 6 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
50	11, 49	sylan 583	. . . . 5 ⊢ ((𝑝 ∈ {{1, 2}} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
51	50	mpteq2dva 5125	. . . 4 ⊢ (𝑝 ∈ {{1, 2}} → (𝑧 ∈ {1, 2} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
52	51	mpteq2ia 5121	. . 3 ⊢ (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
53	10, 52	eqtri 2821	. 2 ⊢ (𝑝 ∈ {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
54	4, 53	eqtri 2821	1 ⊢ (pmTrsp‘{1, 2}) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110  Vcvv 3441   ∖ cdif 3878  ifcif 4425  𝒫 cpw 4497  {csn 4525  {cpr 4527  ∪ cuni 4800   class class class wbr 5030   ↦ cmpt 5110  ‘cfv 6324  2_oc2o 8079   ≈ cen 8489  1c1 10527  2c2 11680  ℕ₀cn0 11885  pmTrspcpmtr 18561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-hash 13687  df-pmtr 18562

This theorem is referenced by:  pmtrprfvalrn  18608