Theorem tpostpos 8198

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 8183	. 2 ⊢ Rel tpos tpos 𝐹
2		relinxp 5771	. 2 ⊢ Rel (𝐹 ∩ (((V × V) ∪ {∅}) × V))
3		relcnv 6071	. . . . . . . . 9 ⊢ Rel ◡dom tpos 𝐹
4		df-rel 5639	. . . . . . . . 9 ⊢ (Rel ◡dom tpos 𝐹 ↔ ◡dom tpos 𝐹 ⊆ (V × V))
5	3, 4	mpbi 230	. . . . . . . 8 ⊢ ◡dom tpos 𝐹 ⊆ (V × V)
6		simpl 482	. . . . . . . 8 ⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) → 𝑤 ∈ ◡dom tpos 𝐹)
7	5, 6	sselid 3933	. . . . . . 7 ⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) → 𝑤 ∈ (V × V))
8		simpr 484	. . . . . . 7 ⊢ ((𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)) → 𝑤 ∈ (V × V))
9		elvv 5707	. . . . . . . . 9 ⊢ (𝑤 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
10		eleq1 2825	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ ◡dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡dom tpos 𝐹))
11		vex 3446	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
12		vex 3446	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
13	11, 12	opelcnv 5838	. . . . . . . . . . . . . 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 4592	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → {𝑤} = {⟨𝑥, 𝑦⟩})
16	15	cnveqd 5832	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ◡{𝑤} = ◡{⟨𝑥, 𝑦⟩})
17	16	unieqd 4878	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ∪ ◡{𝑤} = ∪ ◡{⟨𝑥, 𝑦⟩})
18		opswap 6195	. . . . . . . . . . . . . . 15 ⊢ ∪ ◡{⟨𝑥, 𝑦⟩} = ⟨𝑦, 𝑥⟩
19	17, 18	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ∪ ◡{𝑤} = ⟨𝑦, 𝑥⟩)
20	19	breq1d 5110	. . . . . . . . . . . . 13 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧))
21	14, 20	anbi12d 633	. . . . . . . . . . . 12 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧)))
22		opex 5419	. . . . . . . . . . . . . . 15 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
23		vex 3446	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
24	22, 23	breldm 5865	. . . . . . . . . . . . . 14 ⊢ (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 → ⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹)
25	24	pm4.71ri 560	. . . . . . . . . . . . 13 ⊢ (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ (⟨𝑦, 𝑥⟩ ∈ dom tpos 𝐹 ∧ ⟨𝑦, 𝑥⟩tpos 𝐹𝑧))
26		brtpos 8187	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → (⟨𝑦, 𝑥⟩tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩𝐹𝑧))
27	26	elv 3447	. . . . . . . . . . . . 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 5103	. . . . . . . . . . 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 1934	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
33	9, 32	sylbi 217	. . . . . . . 8 ⊢ (𝑤 ∈ (V × V) → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧))
34		iba 527	. . . . . . . 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 378	. . . . . 6 ⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)))
37		elsni 4599	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ {∅} → 𝑤 = ∅)
38	37	sneqd 4594	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {∅} → {𝑤} = {∅})
39	38	cnveqd 5832	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {∅} → ◡{𝑤} = ◡{∅})
40		cnvsn0 6176	. . . . . . . . . . . . . 14 ⊢ ◡{∅} = ∅
41	39, 40	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ {∅} → ◡{𝑤} = ∅)
42	41	unieqd 4878	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {∅} → ∪ ◡{𝑤} = ∪ ∅)
43		uni0 4893	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
44	42, 43	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {∅} → ∪ ◡{𝑤} = ∅)
45	44	breq1d 5110	. . . . . . . . . 10 ⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ ∅tpos 𝐹𝑧))
46		brtpos0 8185	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
47	46	elv 3447	. . . . . . . . . 10 ⊢ (∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧)
48	45, 47	bitrdi 287	. . . . . . . . 9 ⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ ∅𝐹𝑧))
49	37	breq1d 5110	. . . . . . . . 9 ⊢ (𝑤 ∈ {∅} → (𝑤𝐹𝑧 ↔ ∅𝐹𝑧))
50	48, 49	bitr4d 282	. . . . . . . 8 ⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ 𝑤𝐹𝑧))
51	50	pm5.32i 574	. . . . . . 7 ⊢ ((𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤 ∈ {∅} ∧ 𝑤𝐹𝑧))
52	51	biancomi 462	. . . . . 6 ⊢ ((𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅}))
53	36, 52	orbi12i 915	. . . . 5 ⊢ (((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)) ↔ ((𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅})))
54		andir 1011	. . . . 5 ⊢ (((𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)))
55		andi 1010	. . . . 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 4107	. . . . 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 5681	. . . . . . 7 ⊢ (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ (𝑤 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 ∈ V))
60	23, 59	mpbiran2 711	. . . . . 6 ⊢ (𝑤(((V × V) ∪ {∅}) × V)𝑧 ↔ 𝑤 ∈ ((V × V) ∪ {∅}))
61		elun 4107	. . . . . 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 8184	. . . 4 ⊢ (𝑧 ∈ V → (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)))
66	65	elv 3447	. . 3 ⊢ (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧))
67		brin 5152	. . 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 5748	1 ⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {csn 4582  ⟨cop 4588  ∪ cuni 4865   class class class wbr 5100   × cxp 5630  ^◡ccnv 5631  dom cdm 5632  Rel wrel 5637  tpos ctpos 8177

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-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  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-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508  df-tpos 8178

This theorem is referenced by:  tpostpos2  8199