Theorem reldmtpos 8279

Description: Necessary and sufficient condition for dom tpos 𝐹 to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
reldmtpos	⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)

Proof of Theorem reldmtpos

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

Step	Hyp	Ref	Expression
1		0ex 5264	. . . . 5 ⊢ ∅ ∈ V
2	1	eldm 5889	. . . 4 ⊢ (∅ ∈ dom 𝐹 ↔ ∃𝑦∅𝐹𝑦)
3		brtpos0 8278	. . . . . . 7 ⊢ (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
4	3	elv 3459	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
5		0nelrel0 5715	. . . . . . 7 ⊢ (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹)
6		vex 3458	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	1, 6	breldm 5897	. . . . . . 7 ⊢ (∅tpos 𝐹𝑦 → ∅ ∈ dom tpos 𝐹)
8	5, 7	nsyl3 139	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 → ¬ Rel dom tpos 𝐹)
9	4, 8	sylbir 238	. . . . 5 ⊢ (∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
10	9	exlimiv 1965	. . . 4 ⊢ (∃𝑦∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
11	2, 10	sylbi 220	. . 3 ⊢ (∅ ∈ dom 𝐹 → ¬ Rel dom tpos 𝐹)
12	11	con2i 140	. 2 ⊢ (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom 𝐹)
13		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
14	13	eldm 5889	. . . . 5 ⊢ (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
15		relcnv 6112	. . . . . . . . . . 11 ⊢ Rel ◡dom 𝐹
16		df-rel 5662	. . . . . . . . . . 11 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
17	15, 16	mpbi 233	. . . . . . . . . 10 ⊢ ◡dom 𝐹 ⊆ (V × V)
18	17	sseli 3909	. . . . . . . . 9 ⊢ (𝑥 ∈ ◡dom 𝐹 → 𝑥 ∈ (V × V))
19	18	a1i 11	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ ◡dom 𝐹 → 𝑥 ∈ (V × V)))
20		elsni 4601	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
21	20	breq1d 5116	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
22	1, 6	breldm 5897	. . . . . . . . . . . . 13 ⊢ (∅𝐹𝑦 → ∅ ∈ dom 𝐹)
23	22	pm2.24d 152	. . . . . . . . . . . 12 ⊢ (∅𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V)))
24	4, 23	sylbi 220	. . . . . . . . . . 11 ⊢ (∅tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V)))
25	21, 24	biimtrdi 256	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V))))
26	25	com3l 90	. . . . . . . . 9 ⊢ (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V))))
27	26	impcom 413	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
28		brtpos2 8277	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑥}𝐹𝑦)))
29	6, 28	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑥}𝐹𝑦))
30	29	simplbi 502	. . . . . . . . . 10 ⊢ (𝑥tpos 𝐹𝑦 → 𝑥 ∈ (◡dom 𝐹 ∪ {∅}))
31		elun 4155	. . . . . . . . . 10 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↔ (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
32	30, 31	sylib 221	. . . . . . . . 9 ⊢ (𝑥tpos 𝐹𝑦 → (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
33	32	adantl 487	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
34	19, 27, 33	mpjaod 874	. . . . . . 7 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
35	34	ex 418	. . . . . 6 ⊢ (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦 → 𝑥 ∈ (V × V)))
36	35	exlimdv 1968	. . . . 5 ⊢ (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦 → 𝑥 ∈ (V × V)))
37	14, 36	biimtrid 245	. . . 4 ⊢ (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹 → 𝑥 ∈ (V × V)))
38	37	ssrdv 3919	. . 3 ⊢ (¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ (V × V))
39		df-rel 5662	. . 3 ⊢ (Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ (V × V))
40	38, 39	sylibr 237	. 2 ⊢ (¬ ∅ ∈ dom 𝐹 → Rel dom tpos 𝐹)
41	12, 40	impbii 212	1 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  {csn 4588  ∪ cuni 4870   class class class wbr 5106   × cxp 5653  ^◡ccnv 5654  dom cdm 5655  Rel wrel 5660  tpos ctpos 8270

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-fv 6566  df-tpos 8271

This theorem is used by:  dmtpos  8283  2oppf  50453