Theorem pmtrprfval 19588

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 5414	. . 3 ⊢ {1, 2} ∈ V
2		eqid 2766	. . . 4 ⊢ (pmTrsp‘{1, 2}) = (pmTrsp‘{1, 2})
3	2	pmtrfval 19551	. . 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 11221	. . . . 5 ⊢ 1 ∈ V
6		2nn0 12539	. . . . 5 ⊢ 2 ∈ ℕ0
7		1ne2 12469	. . . . 5 ⊢ 1 ≠ 2
8		pr2pwpr 14536	. . . . 5 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ0 ∧ 1 ≠ 2) → {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} = {{1, 2}})
9	5, 6, 7, 8	mp3an 1490	. . . 4 ⊢ {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} = {{1, 2}}
10	9	mpteq1i 5207	. . 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 4611	. . . . . 6 ⊢ (𝑝 ∈ {{1, 2}} → 𝑝 = {1, 2})
12		eleq2 2855	. . . . . . . . 9 ⊢ (𝑝 = {1, 2} → (𝑧 ∈ 𝑝 ↔ 𝑧 ∈ {1, 2}))
13	12	biimpar 483	. . . . . . . 8 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → 𝑧 ∈ 𝑝)
14	13	iftrued 4500	. . . . . . 7 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = ∪ (𝑝 ∖ {𝑧}))
15		elpri 4618	. . . . . . . . 9 ⊢ (𝑧 ∈ {1, 2} → (𝑧 = 1 ∨ 𝑧 = 2))
16		2ex 12336	. . . . . . . . . . . . 13 ⊢ 2 ∈ V
17	16	unisn 4896	. . . . . . . . . . . 12 ⊢ ∪ {2} = 2
18		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
19		sneq 4604	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 1 → {𝑧} = {1})
20	19	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → {𝑧} = {1})
21	18, 20	difeq12d 4085	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {1}))
22		difprsn1 4773	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({1, 2} ∖ {1}) = {2})
23	7, 22	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({1, 2} ∖ {1}) = {2}
24	21, 23	eqtrdi 2817	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
25	24	unieqd 4890	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = ∪ {2})
26		iftrue 4498	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1 → if(𝑧 = 1, 2, 1) = 2)
27	26	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 2)
28	17, 25, 27	3eqtr4a 2827	. . . . . . . . . . 11 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
29	28	ex 418	. . . . . . . . . 10 ⊢ (𝑧 = 1 → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
30	5	unisn 4896	. . . . . . . . . . . 12 ⊢ ∪ {1} = 1
31		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
32		sneq 4604	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 2 → {𝑧} = {2})
33	32	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → {𝑧} = {2})
34	31, 33	difeq12d 4085	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {2}))
35		difprsn2 4774	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({1, 2} ∖ {2}) = {1})
36	7, 35	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({1, 2} ∖ {2}) = {1}
37	34, 36	eqtrdi 2817	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
38	37	unieqd 4890	. . . . . . . . . . . 12 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = ∪ {1})
39	7	nesymi 3018	. . . . . . . . . . . . . . 15 ⊢ ¬ 2 = 1
40		eqeq1 2770	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 2 → (𝑧 = 1 ↔ 2 = 1))
41	39, 40	mtbiri 330	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 2 → ¬ 𝑧 = 1)
42	41	iffalsed 4503	. . . . . . . . . . . . 13 ⊢ (𝑧 = 2 → if(𝑧 = 1, 2, 1) = 1)
43	42	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 1)
44	30, 38, 43	3eqtr4a 2827	. . . . . . . . . . 11 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
45	44	ex 418	. . . . . . . . . 10 ⊢ (𝑧 = 2 → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
46	29, 45	jaoi 871	. . . . . . . . 9 ⊢ ((𝑧 = 1 ∨ 𝑧 = 2) → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
47	15, 46	syl 18	. . . . . . . 8 ⊢ (𝑧 ∈ {1, 2} → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
48	47	impcom 413	. . . . . . 7 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
49	14, 48	eqtrd 2801	. . . . . 6 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
50	11, 49	sylan 592	. . . . 5 ⊢ ((𝑝 ∈ {{1, 2}} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
51	50	mpteq2dva 5209	. . . 4 ⊢ (𝑝 ∈ {{1, 2}} → (𝑧 ∈ {1, 2} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
52	51	mpteq2ia 5211	. . 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 2789	. 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 2789	1 ⊢ (pmTrsp‘{1, 2}) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  {crab 3419  Vcvv 3458   ∖ cdif 3905  ifcif 4492  𝒫 cpw 4567  {csn 4594  {cpr 4596  ∪ cuni 4877   class class class wbr 5114   ↦ cmpt 5197  ‘cfv 6543  2_oc2o 8456   ≈ cen 8949  1c1 11119  2c2 12313  ℕ₀cn0 12522  pmTrspcpmtr 19542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-dju 9906  df-card 9944  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-n0 12523  df-z 12610  df-uz 12881  df-fz 13554  df-hash 14387  df-pmtr 19543

This theorem is used by:  pmtrprfvalrn  19589