Theorem dftpos4 5901

Description: Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)

Assertion

Ref	Expression
dftpos4	⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))

Distinct variable group:   𝑥,𝐹

Proof of Theorem dftpos4

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

Step	Hyp	Ref	Expression
1		df-tpos 5883	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2		relcnv 4723	. . . . . . 7 ⊢ Rel ◡dom 𝐹
3		df-rel 4370	. . . . . . 7 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
4	2, 3	mpbi 143	. . . . . 6 ⊢ ◡dom 𝐹 ⊆ (V × V)
5		unss1 3141	. . . . . 6 ⊢ (◡dom 𝐹 ⊆ (V × V) → (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
6		resmpt 4676	. . . . . 6 ⊢ ((◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}) → ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
7	4, 5, 6	mp2b 8	. . . . 5 ⊢ ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
8		resss 4653	. . . . 5 ⊢ ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})
9	7, 8	eqsstr3i 3030	. . . 4 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})
10		coss2 4510	. . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})))
11	9, 10	ax-mp 7	. . 3 ⊢ (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
12	1, 11	eqsstri 3029	. 2 ⊢ tpos 𝐹 ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
13		relco 4839	. . 3 ⊢ Rel (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
14		vex 2604	. . . . 5 ⊢ 𝑦 ∈ V
15		vex 2604	. . . . 5 ⊢ 𝑧 ∈ V
16	14, 15	opelco 4525	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ↔ ∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ∧ 𝑤𝐹𝑧))
17		vex 2604	. . . . . . . . 9 ⊢ 𝑤 ∈ V
18		eleq1 2141	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ((V × V) ∪ {∅}) ↔ 𝑦 ∈ ((V × V) ∪ {∅})))
19		sneq 3409	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
20	19	cnveqd 4529	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ◡{𝑥} = ◡{𝑦})
21	20	unieqd 3612	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ∪ ◡{𝑥} = ∪ ◡{𝑦})
22	21	eqeq2d 2092	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑧 = ∪ ◡{𝑥} ↔ 𝑧 = ∪ ◡{𝑦}))
23	18, 22	anbi12d 456	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = ∪ ◡{𝑥}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = ∪ ◡{𝑦})))
24		eqeq1 2087	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑧 = ∪ ◡{𝑦} ↔ 𝑤 = ∪ ◡{𝑦}))
25	24	anbi2d 451	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = ∪ ◡{𝑦}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦})))
26		df-mpt 3841	. . . . . . . . 9 ⊢ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = ∪ ◡{𝑥})}
27	14, 17, 23, 25, 26	brab 4027	. . . . . . . 8 ⊢ (𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}))
28		simplr 496	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 = ∪ ◡{𝑦})
29	17, 15	breldm 4557	. . . . . . . . . . . . 13 ⊢ (𝑤𝐹𝑧 → 𝑤 ∈ dom 𝐹)
30	29	adantl 271	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 ∈ dom 𝐹)
31	28, 30	eqeltrrd 2156	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → ∪ ◡{𝑦} ∈ dom 𝐹)
32		elvv 4420	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
33		opswapg 4827	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → ∪ ◡{⟨𝑧, 𝑤⟩} = ⟨𝑤, 𝑧⟩)
34	15, 17, 33	mp2an 416	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ ◡{⟨𝑧, 𝑤⟩} = ⟨𝑤, 𝑧⟩
35	34	eleq1i 2144	. . . . . . . . . . . . . . . . 17 ⊢ (∪ ◡{⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
36	15, 17	opelcnv 4535	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
37	35, 36	bitr4i 185	. . . . . . . . . . . . . . . 16 ⊢ (∪ ◡{⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ ◡dom 𝐹)
38		sneq 3409	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
39	38	cnveqd 4529	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ◡{𝑦} = ◡{⟨𝑧, 𝑤⟩})
40	39	unieqd 3612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∪ ◡{𝑦} = ∪ ◡{⟨𝑧, 𝑤⟩})
41	40	eleq1d 2147	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (∪ ◡{𝑦} ∈ dom 𝐹 ↔ ∪ ◡{⟨𝑧, 𝑤⟩} ∈ dom 𝐹))
42		eleq1 2141	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦 ∈ ◡dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ ◡dom 𝐹))
43	41, 42	bibi12d 233	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ((∪ ◡{𝑦} ∈ dom 𝐹 ↔ 𝑦 ∈ ◡dom 𝐹) ↔ (∪ ◡{⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ ◡dom 𝐹)))
44	37, 43	mpbiri 166	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (∪ ◡{𝑦} ∈ dom 𝐹 ↔ 𝑦 ∈ ◡dom 𝐹))
45	44	exlimivv 1817	. . . . . . . . . . . . . 14 ⊢ (∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (∪ ◡{𝑦} ∈ dom 𝐹 ↔ 𝑦 ∈ ◡dom 𝐹))
46	32, 45	sylbi 119	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (V × V) → (∪ ◡{𝑦} ∈ dom 𝐹 ↔ 𝑦 ∈ ◡dom 𝐹))
47	46	biimpcd 157	. . . . . . . . . . . 12 ⊢ (∪ ◡{𝑦} ∈ dom 𝐹 → (𝑦 ∈ (V × V) → 𝑦 ∈ ◡dom 𝐹))
48		elun1 3139	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ◡dom 𝐹 → 𝑦 ∈ (◡dom 𝐹 ∪ {∅}))
49	47, 48	syl6 33	. . . . . . . . . . 11 ⊢ (∪ ◡{𝑦} ∈ dom 𝐹 → (𝑦 ∈ (V × V) → 𝑦 ∈ (◡dom 𝐹 ∪ {∅})))
50	31, 49	syl 14	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (V × V) → 𝑦 ∈ (◡dom 𝐹 ∪ {∅})))
51		elun2 3140	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {∅} → 𝑦 ∈ (◡dom 𝐹 ∪ {∅}))
52	51	a1i 9	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ {∅} → 𝑦 ∈ (◡dom 𝐹 ∪ {∅})))
53		simpll 495	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ ((V × V) ∪ {∅}))
54		elun 3113	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((V × V) ∪ {∅}) ↔ (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
55	53, 54	sylib 120	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
56	50, 52, 55	mpjaod 670	. . . . . . . . 9 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ (◡dom 𝐹 ∪ {∅}))
57		simpr 108	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤𝐹𝑧)
58	28, 57	eqbrtrrd 3807	. . . . . . . . 9 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → ∪ ◡{𝑦}𝐹𝑧)
59	56, 58	jca 300	. . . . . . . 8 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑦}𝐹𝑧))
60	27, 59	sylanb 278	. . . . . . 7 ⊢ ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑦}𝐹𝑧))
61		brtpos2 5889	. . . . . . . 8 ⊢ (𝑧 ∈ V → (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑦}𝐹𝑧)))
62	15, 61	ax-mp 7	. . . . . . 7 ⊢ (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑦}𝐹𝑧))
63	60, 62	sylibr 132	. . . . . 6 ⊢ ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ∧ 𝑤𝐹𝑧) → 𝑦tpos 𝐹𝑧)
64		df-br 3786	. . . . . 6 ⊢ (𝑦tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
65	63, 64	sylib 120	. . . . 5 ⊢ ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ∧ 𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
66	65	exlimiv 1529	. . . 4 ⊢ (∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ∧ 𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
67	16, 66	sylbi 119	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
68	13, 67	relssi 4449	. 2 ⊢ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ tpos 𝐹
69	12, 68	eqssi 3015	1 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661   = wceq 1284  ∃wex 1421   ∈ wcel 1433  Vcvv 2601   ∪ cun 2971   ⊆ wss 2973  ∅c0 3251  {csn 3398  ⟨cop 3401  ∪ cuni 3601   class class class wbr 3785   ↦ cmpt 3839   × cxp 4361  ^◡ccnv 4362  dom cdm 4363   ↾ cres 4365   ∘ ccom 4367  Rel wrel 4368  tpos ctpos 5882

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930  df-tpos 5883

This theorem is referenced by:  tposco  5913  nftpos  5917