Theorem dftpos4 8227

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 8208	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2		relcnv 6078	. . . . . . 7 ⊢ Rel ◡dom 𝐹
3		df-rel 5648	. . . . . . 7 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
4	2, 3	mpbi 230	. . . . . 6 ⊢ ◡dom 𝐹 ⊆ (V × V)
5		unss1 4151	. . . . . 6 ⊢ (◡dom 𝐹 ⊆ (V × V) → (◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}))
6		resmpt 6011	. . . . . 6 ⊢ ((◡dom 𝐹 ∪ {∅}) ⊆ ((V × V) ∪ {∅}) → ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
7	4, 5, 6	mp2b 10	. . . . 5 ⊢ ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
8		resss 5975	. . . . 5 ⊢ ((𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ (◡dom 𝐹 ∪ {∅})) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})
9	7, 8	eqsstrri 3997	. . . 4 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})
10		coss2 5823	. . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) → (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})))
11	9, 10	ax-mp 5	. . 3 ⊢ (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
12	1, 11	eqsstri 3996	. 2 ⊢ tpos 𝐹 ⊆ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
13		relco 6082	. . 3 ⊢ Rel (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
14		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
15		vex 3454	. . . . 5 ⊢ 𝑧 ∈ V
16	14, 15	opelco 5838	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ↔ ∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ∧ 𝑤𝐹𝑧))
17		vex 3454	. . . . . . . . 9 ⊢ 𝑤 ∈ V
18		eleq1 2817	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ((V × V) ∪ {∅}) ↔ 𝑦 ∈ ((V × V) ∪ {∅})))
19		sneq 4602	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
20	19	cnveqd 5842	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ◡{𝑥} = ◡{𝑦})
21	20	unieqd 4887	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ∪ ◡{𝑥} = ∪ ◡{𝑦})
22	21	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑧 = ∪ ◡{𝑥} ↔ 𝑧 = ∪ ◡{𝑦}))
23	18, 22	anbi12d 632	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = ∪ ◡{𝑥}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = ∪ ◡{𝑦})))
24		eqeq1 2734	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑧 = ∪ ◡{𝑦} ↔ 𝑤 = ∪ ◡{𝑦}))
25	24	anbi2d 630	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = ∪ ◡{𝑦}) ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦})))
26		df-mpt 5192	. . . . . . . . 9 ⊢ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ((V × V) ∪ {∅}) ∧ 𝑧 = ∪ ◡{𝑥})}
27	14, 17, 23, 25, 26	brab 5506	. . . . . . . 8 ⊢ (𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ↔ (𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}))
28		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 = ∪ ◡{𝑦})
29	17, 15	breldm 5875	. . . . . . . . . . . . 13 ⊢ (𝑤𝐹𝑧 → 𝑤 ∈ dom 𝐹)
30	29	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤 ∈ dom 𝐹)
31	28, 30	eqeltrrd 2830	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → ∪ ◡{𝑦} ∈ dom 𝐹)
32		elvv 5716	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
33		opswap 6205	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ ◡{⟨𝑧, 𝑤⟩} = ⟨𝑤, 𝑧⟩
34	33	eleq1i 2820	. . . . . . . . . . . . . . . . 17 ⊢ (∪ ◡{⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
35	15, 17	opelcnv 5848	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡dom 𝐹 ↔ ⟨𝑤, 𝑧⟩ ∈ dom 𝐹)
36	34, 35	bitr4i 278	. . . . . . . . . . . . . . . 16 ⊢ (∪ ◡{⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ ◡dom 𝐹)
37		sneq 4602	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → {𝑦} = {⟨𝑧, 𝑤⟩})
38	37	cnveqd 5842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ◡{𝑦} = ◡{⟨𝑧, 𝑤⟩})
39	38	unieqd 4887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∪ ◡{𝑦} = ∪ ◡{⟨𝑧, 𝑤⟩})
40	39	eleq1d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (∪ ◡{𝑦} ∈ dom 𝐹 ↔ ∪ ◡{⟨𝑧, 𝑤⟩} ∈ dom 𝐹))
41		eleq1 2817	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦 ∈ ◡dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ ◡dom 𝐹))
42	40, 41	bibi12d 345	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ((∪ ◡{𝑦} ∈ dom 𝐹 ↔ 𝑦 ∈ ◡dom 𝐹) ↔ (∪ ◡{⟨𝑧, 𝑤⟩} ∈ dom 𝐹 ↔ ⟨𝑧, 𝑤⟩ ∈ ◡dom 𝐹)))
43	36, 42	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (∪ ◡{𝑦} ∈ dom 𝐹 ↔ 𝑦 ∈ ◡dom 𝐹))
44	43	exlimivv 1932	. . . . . . . . . . . . . 14 ⊢ (∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (∪ ◡{𝑦} ∈ dom 𝐹 ↔ 𝑦 ∈ ◡dom 𝐹))
45	32, 44	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (V × V) → (∪ ◡{𝑦} ∈ dom 𝐹 ↔ 𝑦 ∈ ◡dom 𝐹))
46	45	biimpcd 249	. . . . . . . . . . . 12 ⊢ (∪ ◡{𝑦} ∈ dom 𝐹 → (𝑦 ∈ (V × V) → 𝑦 ∈ ◡dom 𝐹))
47		elun1 4148	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ◡dom 𝐹 → 𝑦 ∈ (◡dom 𝐹 ∪ {∅}))
48	46, 47	syl6 35	. . . . . . . . . . 11 ⊢ (∪ ◡{𝑦} ∈ dom 𝐹 → (𝑦 ∈ (V × V) → 𝑦 ∈ (◡dom 𝐹 ∪ {∅})))
49	31, 48	syl 17	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (V × V) → 𝑦 ∈ (◡dom 𝐹 ∪ {∅})))
50		elun2 4149	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {∅} → 𝑦 ∈ (◡dom 𝐹 ∪ {∅}))
51	50	a1i 11	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ {∅} → 𝑦 ∈ (◡dom 𝐹 ∪ {∅})))
52		simpll 766	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ ((V × V) ∪ {∅}))
53		elun 4119	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((V × V) ∪ {∅}) ↔ (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
54	52, 53	sylib 218	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (V × V) ∨ 𝑦 ∈ {∅}))
55	49, 51, 54	mpjaod 860	. . . . . . . . 9 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → 𝑦 ∈ (◡dom 𝐹 ∪ {∅}))
56		simpr 484	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → 𝑤𝐹𝑧)
57	28, 56	eqbrtrrd 5134	. . . . . . . . 9 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → ∪ ◡{𝑦}𝐹𝑧)
58	55, 57	jca 511	. . . . . . . 8 ⊢ (((𝑦 ∈ ((V × V) ∪ {∅}) ∧ 𝑤 = ∪ ◡{𝑦}) ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑦}𝐹𝑧))
59	27, 58	sylanb 581	. . . . . . 7 ⊢ ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ∧ 𝑤𝐹𝑧) → (𝑦 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑦}𝐹𝑧))
60		brtpos2 8214	. . . . . . . 8 ⊢ (𝑧 ∈ V → (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑦}𝐹𝑧)))
61	15, 60	ax-mp 5	. . . . . . 7 ⊢ (𝑦tpos 𝐹𝑧 ↔ (𝑦 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑦}𝐹𝑧))
62	59, 61	sylibr 234	. . . . . 6 ⊢ ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ∧ 𝑤𝐹𝑧) → 𝑦tpos 𝐹𝑧)
63		df-br 5111	. . . . . 6 ⊢ (𝑦tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
64	62, 63	sylib 218	. . . . 5 ⊢ ((𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ∧ 𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
65	64	exlimiv 1930	. . . 4 ⊢ (∃𝑤(𝑦(𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})𝑤 ∧ 𝑤𝐹𝑧) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
66	16, 65	sylbi 217	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) → ⟨𝑦, 𝑧⟩ ∈ tpos 𝐹)
67	13, 66	relssi 5753	. 2 ⊢ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ tpos 𝐹
68	12, 67	eqssi 3966	1 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  ∅c0 4299  {csn 4592  ⟨cop 4598  ∪ cuni 4874   class class class wbr 5110   ↦ cmpt 5191   × cxp 5639  ^◡ccnv 5640  dom cdm 5641   ↾ cres 5643   ∘ ccom 5645  Rel wrel 5646  tpos ctpos 8207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522  df-tpos 8208

This theorem is referenced by:  tposco  8239  nftpos  8243  oftpos  22346