Theorem tpostpos 8226

Description: Value of the double transposition for a general class 𝐹. (Contributed by Mario Carneiro, 16-Sep-2015.)

Assertion

Ref	Expression
tpostpos	⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))

Proof of Theorem tpostpos

Dummy variables 𝑥 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reltpos 8211	. 2 ⊢ Rel tpos tpos 𝐹
2		relinxp 5812	. 2 ⊢ Rel (𝐹 ∩ (((V × V) ∪ {∅}) × V))
3		relcnv 6100	. . . . . . . . 9 ⊢ Rel ◡dom tpos 𝐹
4		df-rel 5682	. . . . . . . . 9 ⊢ (Rel ◡dom tpos 𝐹 ↔ ◡dom tpos 𝐹 ⊆ (V × V))
5	3, 4	mpbi 229	. . . . . . . 8 ⊢ ◡dom tpos 𝐹 ⊆ (V × V)
6		simpl 484	. . . . . . . 8 ⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) → 𝑤 ∈ ◡dom tpos 𝐹)
7	5, 6	sselid 3979	. . . . . . 7 ⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) → 𝑤 ∈ (V × V))
8		simpr 486	. . . . . . 7 ⊢ ((𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)) → 𝑤 ∈ (V × V))
9		elvv 5748	. . . . . . . . 9 ⊢ (𝑤 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
10		eleq1 2822	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ ◡dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡dom tpos 𝐹))
11		vex 3479	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
12		vex 3479	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
13	11, 12	opelcnv 5879	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹)
14	10, 13	bitrdi 287	. . . . . . . . . . . . 13 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ ◡dom tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹))
15		sneq 4637	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
16	15	cnveqd 5873	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ◡{𝑤} = ◡{⟨𝑥, 𝑦⟩})
17	16	unieqd 4921	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ∪ ◡{𝑤} = ∪ ◡{⟨𝑥, 𝑦⟩})
18		opswap 6225	. . . . . . . . . . . . . . 15 ⊢ ∪ ◡{⟨𝑥, 𝑦⟩} = ⟨𝑦, 𝑥⟩
19	17, 18	eqtrdi 2789	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ∪ ◡{𝑤} = ⟨𝑦, 𝑥⟩)
20	19	breq1d 5157	. . . . . . . . . . . . 13 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧))
21	14, 20	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧)))
22		opex 5463	. . . . . . . . . . . . . . 15 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
23		vex 3479	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
24	22, 23	breldm 5906	. . . . . . . . . . . . . 14 ⊢ (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 → ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹)
25	24	pm4.71ri 562	. . . . . . . . . . . . 13 ⊢ (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ (⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧))
26		brtpos 8215	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩𝐹𝑧))
27	26	elv 3481	. . . . . . . . . . . . 13 ⊢ (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩𝐹𝑧)
28	25, 27	bitr3i 277	. . . . . . . . . . . 12 ⊢ ((⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧) ↔ ⟨𝑥, 𝑦⟩𝐹𝑧)
29	21, 28	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ ⟨𝑥, 𝑦⟩𝐹𝑧))
30		breq1 5150	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩𝐹𝑧))
31	29, 30	bitr4d 282	. . . . . . . . . 10 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
32	31	exlimivv 1936	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
33	9, 32	sylbi 216	. . . . . . . 8 ⊢ (𝑤 ∈ (V × V) → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
34		iba 529	. . . . . . . 8 ⊢ (𝑤 ∈ (V × V) → (𝑤𝐹𝑧 ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V))))
35	33, 34	bitrd 279	. . . . . . 7 ⊢ (𝑤 ∈ (V × V) → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V))))
36	7, 8, 35	pm5.21nii 380	. . . . . 6 ⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)))
37		elsni 4644	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ {∅} → 𝑤 = ∅)
38	37	sneqd 4639	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {∅} → {𝑤} = {∅})
39	38	cnveqd 5873	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {∅} → ◡{𝑤} = ◡{∅})
40		cnvsn0 6206	. . . . . . . . . . . . . 14 ⊢ ◡{∅} = ∅
41	39, 40	eqtrdi 2789	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ {∅} → ◡{𝑤} = ∅)
42	41	unieqd 4921	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {∅} → ∪ ◡{𝑤} = ∪ ∅)
43		uni0 4938	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
44	42, 43	eqtrdi 2789	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {∅} → ∪ ◡{𝑤} = ∅)
45	44	breq1d 5157	. . . . . . . . . 10 ⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ ∅tpos 𝐹𝑧))
46		brtpos0 8213	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
47	46	elv 3481	. . . . . . . . . 10 ⊢ (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧)
48	45, 47	bitrdi 287	. . . . . . . . 9 ⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
49	37	breq1d 5157	. . . . . . . . 9 ⊢ (𝑤 ∈ {∅} → (𝑤𝐹𝑧 ↔ ∅𝐹𝑧))
50	48, 49	bitr4d 282	. . . . . . . 8 ⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ 𝑤𝐹𝑧))
51	50	pm5.32i 576	. . . . . . 7 ⊢ ((𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤 ∈ {∅} ∧ 𝑤𝐹𝑧))
52	51	biancomi 464	. . . . . 6 ⊢ ((𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅}))
53	36, 52	orbi12i 914	. . . . 5 ⊢ (((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)) ↔ ((𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅})))
54		andir 1008	. . . . 5 ⊢ (((𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)))
55		andi 1007	. . . . 5 ⊢ ((𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})) ↔ ((𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅})))
56	53, 54, 55	3bitr4i 303	. . . 4 ⊢ (((𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})))
57		elun 4147	. . . . 5 ⊢ (𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ↔ (𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅}))
58	57	anbi1i 625	. . . 4 ⊢ ((𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ ((𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧))
59		brxp 5723	. . . . . . 7 ⊢ (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ (𝑤 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 ∈ V))
60	23, 59	mpbiran2 709	. . . . . 6 ⊢ (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ 𝑤 ∈ ((V × V) ∪ {∅}))
61		elun 4147	. . . . . 6 ⊢ (𝑤 ∈ ((V × V) ∪ {∅}) ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))
62	60, 61	bitri 275	. . . . 5 ⊢ (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))
63	62	anbi2i 624	. . . 4 ⊢ ((𝑤𝐹𝑧 ∧ 𝑤(((V × V) ∪ {∅}) × V)𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})))
64	56, 58, 63	3bitr4i 303	. . 3 ⊢ ((𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤(((V × V) ∪ {∅}) × V)𝑧))
65		brtpos2 8212	. . . 4 ⊢ (𝑧 ∈ V → (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)))
66	65	elv 3481	. . 3 ⊢ (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧))
67		brin 5199	. . 3 ⊢ (𝑤(𝐹 ∩ (((V × V) ∪ {∅}) × V))𝑧 ↔ (𝑤𝐹𝑧 ∧ 𝑤(((V × V) ∪ {∅}) × V)𝑧))
68	64, 66, 67	3bitr4i 303	. 2 ⊢ (𝑤tpos tpos 𝐹𝑧 ↔ 𝑤(𝐹 ∩ (((V × V) ∪ {∅}) × V))𝑧)
69	1, 2, 68	eqbrriv 5789	1 ⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Vcvv 3475   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  {csn 4627  ⟨cop 4633  ∪ cuni 4907   class class class wbr 5147   × cxp 5673  ^◡ccnv 5674  dom cdm 5675  Rel wrel 5680  tpos ctpos 8205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-fv 6548  df-tpos 8206

This theorem is referenced by:  tpostpos2  8227