Theorem pmtrprfval 19558

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 5411	. . 3 ⊢ {1, 2} ∈ V
2		eqid 2763	. . . 4 ⊢ (pmTrsp‘{1, 2}) = (pmTrsp‘{1, 2})
3	2	pmtrfval 19521	. . 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 11204	. . . . 5 ⊢ 1 ∈ V
6		2nn0 12522	. . . . 5 ⊢ 2 ∈ ℕ0
7		1ne2 12452	. . . . 5 ⊢ 1 ≠ 2
8		pr2pwpr 14518	. . . . 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 5203	. . 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 4607	. . . . . 6 ⊢ (𝑝 ∈ {{1, 2}} → 𝑝 = {1, 2})
12		eleq2 2852	. . . . . . . . 9 ⊢ (𝑝 = {1, 2} → (𝑧 ∈ 𝑝 ↔ 𝑧 ∈ {1, 2}))
13	12	biimpar 482	. . . . . . . 8 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → 𝑧 ∈ 𝑝)
14	13	iftrued 4496	. . . . . . 7 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = ∪ (𝑝 ∖ {𝑧}))
15		elpri 4614	. . . . . . . . 9 ⊢ (𝑧 ∈ {1, 2} → (𝑧 = 1 ∨ 𝑧 = 2))
16		2ex 12319	. . . . . . . . . . . . 13 ⊢ 2 ∈ V
17	16	unisn 4892	. . . . . . . . . . . 12 ⊢ ∪ {2} = 2
18		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
19		sneq 4600	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 1 → {𝑧} = {1})
20	19	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → {𝑧} = {1})
21	18, 20	difeq12d 4083	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {1}))
22		difprsn1 4769	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({1, 2} ∖ {1}) = {2})
23	7, 22	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({1, 2} ∖ {1}) = {2}
24	21, 23	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
25	24	unieqd 4886	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = ∪ {2})
26		iftrue 4494	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1 → if(𝑧 = 1, 2, 1) = 2)
27	26	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 2)
28	17, 25, 27	3eqtr4a 2824	. . . . . . . . . . 11 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
29	28	ex 417	. . . . . . . . . 10 ⊢ (𝑧 = 1 → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
30	5	unisn 4892	. . . . . . . . . . . 12 ⊢ ∪ {1} = 1
31		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
32		sneq 4600	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 2 → {𝑧} = {2})
33	32	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → {𝑧} = {2})
34	31, 33	difeq12d 4083	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {2}))
35		difprsn2 4770	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({1, 2} ∖ {2}) = {1})
36	7, 35	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({1, 2} ∖ {2}) = {1}
37	34, 36	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
38	37	unieqd 4886	. . . . . . . . . . . 12 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = ∪ {1})
39	7	nesymi 3015	. . . . . . . . . . . . . . 15 ⊢ ¬ 2 = 1
40		eqeq1 2767	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 2 → (𝑧 = 1 ↔ 2 = 1))
41	39, 40	mtbiri 330	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 2 → ¬ 𝑧 = 1)
42	41	iffalsed 4499	. . . . . . . . . . . . 13 ⊢ (𝑧 = 2 → if(𝑧 = 1, 2, 1) = 1)
43	42	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 1)
44	30, 38, 43	3eqtr4a 2824	. . . . . . . . . . 11 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
45	44	ex 417	. . . . . . . . . 10 ⊢ (𝑧 = 2 → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
46	29, 45	jaoi 870	. . . . . . . . 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 412	. . . . . . 7 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
49	14, 48	eqtrd 2798	. . . . . 6 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
50	11, 49	sylan 591	. . . . 5 ⊢ ((𝑝 ∈ {{1, 2}} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
51	50	mpteq2dva 5205	. . . 4 ⊢ (𝑝 ∈ {{1, 2}} → (𝑧 ∈ {1, 2} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
52	51	mpteq2ia 5207	. . 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 2786	. 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 2786	1 ⊢ (pmTrsp‘{1, 2}) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  {crab 3416  Vcvv 3455   ∖ cdif 3903  ifcif 4488  𝒫 cpw 4563  {csn 4590  {cpr 4592  ∪ cuni 4873   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6538  2_oc2o 8448   ≈ cen 8941  1c1 11102  2c2 12296  ℕ₀cn0 12505  pmTrspcpmtr 19512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-hash 14369  df-pmtr 19513

This theorem is referenced by:  pmtrprfvalrn  19559