Theorem pmtrprfval 19706

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 5395	. . 3 ⊢ {1, 2} ∈ V
2		eqid 2760	. . . 4 ⊢ (pmTrsp‘{1, 2}) = (pmTrsp‘{1, 2})
3	2	pmtrfval 19669	. . 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 11306	. . . . 5 ⊢ 1 ∈ V
6		2nn0 12626	. . . . 5 ⊢ 2 ∈ ℕ0
7		1ne2 12556	. . . . 5 ⊢ 1 ≠ 2
8		pr2pwpr 14627	. . . . 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 5195	. . 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 4597	. . . . . 6 ⊢ (𝑝 ∈ {{1, 2}} → 𝑝 = {1, 2})
12		eleq2 2849	. . . . . . . . 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 4489	. . . . . . 7 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = ∪ (𝑝 ∖ {𝑧}))
15		elpri 4604	. . . . . . . . 9 ⊢ (𝑧 ∈ {1, 2} → (𝑧 = 1 ∨ 𝑧 = 2))
16		2ex 12423	. . . . . . . . . . . . 13 ⊢ 2 ∈ V
17	16	unisn 4885	. . . . . . . . . . . 12 ⊢ ∪ {2} = 2
18		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
19		sneq 4590	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 1 → {𝑧} = {1})
20	19	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → {𝑧} = {1})
21	18, 20	difeq12d 4064	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {1}))
22		difprsn1 4759	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({1, 2} ∖ {1}) = {2})
23	7, 22	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({1, 2} ∖ {1}) = {2}
24	21, 23	eqtrdi 2811	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
25	24	unieqd 4879	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = ∪ {2})
26		iftrue 4487	. . . . . . . . . . . . 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 2821	. . . . . . . . . . 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 4885	. . . . . . . . . . . 12 ⊢ ∪ {1} = 1
31		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
32		sneq 4590	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 2 → {𝑧} = {2})
33	32	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → {𝑧} = {2})
34	31, 33	difeq12d 4064	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {2}))
35		difprsn2 4760	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({1, 2} ∖ {2}) = {1})
36	7, 35	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({1, 2} ∖ {2}) = {1}
37	34, 36	eqtrdi 2811	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
38	37	unieqd 4879	. . . . . . . . . . . 12 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = ∪ {1})
39	7	nesymi 3012	. . . . . . . . . . . . . . 15 ⊢ ¬ 2 = 1
40		eqeq1 2764	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 2 → (𝑧 = 1 ↔ 2 = 1))
41	39, 40	mtbiri 330	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 2 → ¬ 𝑧 = 1)
42	41	iffalsed 4492	. . . . . . . . . . . . 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 2821	. . . . . . . . . . 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 2795	. . . . . 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 5197	. . . 4 ⊢ (𝑝 ∈ {{1, 2}} → (𝑧 ∈ {1, 2} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
52	51	mpteq2ia 5199	. . 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 2783	. 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 2783	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 1569   ∈ wcel 2144   ≠ wne 2955  {crab 3412  Vcvv 3450   ∖ cdif 4045  ifcif 4481  𝒫 cpw 4556  {csn 4584  {cpr 4586  ∪ cuni 4866   class class class wbr 5102   ↦ cmpt 5185  ‘cfv 6538  2_oc2o 8473   ≈ cen 8973  1c1 11204  2c2 12400  ℕ₀cn0 12609  pmTrspcpmtr 19660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-oadd 8483  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-dju 9985  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-n0 12610  df-z 12697  df-uz 12969  df-fz 13643  df-hash 14478  df-pmtr 19661

This theorem is used by:  pmtrprfvalrn  19707