Theorem pmtrprfval 19428

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 5384	. . 3 ⊢ {1, 2} ∈ V
2		eqid 2737	. . . 4 ⊢ (pmTrsp‘{1, 2}) = (pmTrsp‘{1, 2})
3	2	pmtrfval 19391	. . 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 11140	. . . . 5 ⊢ 1 ∈ V
6		2nn0 12430	. . . . 5 ⊢ 2 ∈ ℕ0
7		1ne2 12360	. . . . 5 ⊢ 1 ≠ 2
8		pr2pwpr 14414	. . . . 5 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ0 ∧ 1 ≠ 2) → {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} = {{1, 2}})
9	5, 6, 7, 8	mp3an 1464	. . . 4 ⊢ {𝑡 ∈ 𝒫 {1, 2} ∣ 𝑡 ≈ 2o} = {{1, 2}}
10	9	mpteq1i 5191	. . 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 4599	. . . . . 6 ⊢ (𝑝 ∈ {{1, 2}} → 𝑝 = {1, 2})
12		eleq2 2826	. . . . . . . . 9 ⊢ (𝑝 = {1, 2} → (𝑧 ∈ 𝑝 ↔ 𝑧 ∈ {1, 2}))
13	12	biimpar 477	. . . . . . . 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 4606	. . . . . . . . 9 ⊢ (𝑧 ∈ {1, 2} → (𝑧 = 1 ∨ 𝑧 = 2))
16		2ex 12234	. . . . . . . . . . . . 13 ⊢ 2 ∈ V
17	16	unisn 4884	. . . . . . . . . . . 12 ⊢ ∪ {2} = 2
18		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
19		sneq 4592	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 1 → {𝑧} = {1})
20	19	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → {𝑧} = {1})
21	18, 20	difeq12d 4081	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {1}))
22		difprsn1 4758	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({1, 2} ∖ {1}) = {2})
23	7, 22	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({1, 2} ∖ {1}) = {2}
24	21, 23	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {2})
25	24	unieqd 4878	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = ∪ {2})
26		iftrue 4487	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1 → if(𝑧 = 1, 2, 1) = 2)
27	26	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 2)
28	17, 25, 27	3eqtr4a 2798	. . . . . . . . . . 11 ⊢ ((𝑧 = 1 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
29	28	ex 412	. . . . . . . . . 10 ⊢ (𝑧 = 1 → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
30	5	unisn 4884	. . . . . . . . . . . 12 ⊢ ∪ {1} = 1
31		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → 𝑝 = {1, 2})
32		sneq 4592	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 2 → {𝑧} = {2})
33	32	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → {𝑧} = {2})
34	31, 33	difeq12d 4081	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = ({1, 2} ∖ {2}))
35		difprsn2 4759	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → ({1, 2} ∖ {2}) = {1})
36	7, 35	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ({1, 2} ∖ {2}) = {1}
37	34, 36	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → (𝑝 ∖ {𝑧}) = {1})
38	37	unieqd 4878	. . . . . . . . . . . 12 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = ∪ {1})
39	7	nesymi 2990	. . . . . . . . . . . . . . 15 ⊢ ¬ 2 = 1
40		eqeq1 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 2 → (𝑧 = 1 ↔ 2 = 1))
41	39, 40	mtbiri 327	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 2 → ¬ 𝑧 = 1)
42	41	iffalsed 4492	. . . . . . . . . . . . 13 ⊢ (𝑧 = 2 → if(𝑧 = 1, 2, 1) = 1)
43	42	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → if(𝑧 = 1, 2, 1) = 1)
44	30, 38, 43	3eqtr4a 2798	. . . . . . . . . . 11 ⊢ ((𝑧 = 2 ∧ 𝑝 = {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
45	44	ex 412	. . . . . . . . . 10 ⊢ (𝑧 = 2 → (𝑝 = {1, 2} → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1)))
46	29, 45	jaoi 858	. . . . . . . . 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 407	. . . . . . 7 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → ∪ (𝑝 ∖ {𝑧}) = if(𝑧 = 1, 2, 1))
49	14, 48	eqtrd 2772	. . . . . 6 ⊢ ((𝑝 = {1, 2} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
50	11, 49	sylan 581	. . . . 5 ⊢ ((𝑝 ∈ {{1, 2}} ∧ 𝑧 ∈ {1, 2}) → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 = 1, 2, 1))
51	50	mpteq2dva 5193	. . . 4 ⊢ (𝑝 ∈ {{1, 2}} → (𝑧 ∈ {1, 2} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
52	51	mpteq2ia 5195	. . 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 2760	. 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 2760	1 ⊢ (pmTrsp‘{1, 2}) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3401  Vcvv 3442   ∖ cdif 3900  ifcif 4481  𝒫 cpw 4556  {csn 4582  {cpr 4584  ∪ cuni 4865   class class class wbr 5100   ↦ cmpt 5181  ‘cfv 6500  2_oc2o 8401   ≈ cen 8892  1c1 11039  2c2 12212  ℕ₀cn0 12413  pmTrspcpmtr 19382

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-hash 14266  df-pmtr 19383

This theorem is referenced by:  pmtrprfvalrn  19429