Theorem reldmtpos 8216

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 5265	. . . . 5 ⊢ ∅ ∈ V
2	1	eldm 5867	. . . 4 ⊢ (∅ ∈ dom 𝐹 ↔ ∃𝑦∅𝐹𝑦)
3		brtpos0 8215	. . . . . . 7 ⊢ (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
4	3	elv 3455	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
5		0nelrel0 5701	. . . . . . 7 ⊢ (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹)
6		vex 3454	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	1, 6	breldm 5875	. . . . . . 7 ⊢ (∅tpos 𝐹𝑦 → ∅ ∈ dom tpos 𝐹)
8	5, 7	nsyl3 138	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 → ¬ Rel dom tpos 𝐹)
9	4, 8	sylbir 235	. . . . 5 ⊢ (∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
10	9	exlimiv 1930	. . . 4 ⊢ (∃𝑦∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
11	2, 10	sylbi 217	. . 3 ⊢ (∅ ∈ dom 𝐹 → ¬ Rel dom tpos 𝐹)
12	11	con2i 139	. 2 ⊢ (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom 𝐹)
13		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
14	13	eldm 5867	. . . . 5 ⊢ (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
15		relcnv 6078	. . . . . . . . . . 11 ⊢ Rel ◡dom 𝐹
16		df-rel 5648	. . . . . . . . . . 11 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
17	15, 16	mpbi 230	. . . . . . . . . 10 ⊢ ◡dom 𝐹 ⊆ (V × V)
18	17	sseli 3945	. . . . . . . . 9 ⊢ (𝑥 ∈ ◡dom 𝐹 → 𝑥 ∈ (V × V))
19	18	a1i 11	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ ◡dom 𝐹 → 𝑥 ∈ (V × V)))
20		elsni 4609	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
21	20	breq1d 5120	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
22	1, 6	breldm 5875	. . . . . . . . . . . . 13 ⊢ (∅𝐹𝑦 → ∅ ∈ dom 𝐹)
23	22	pm2.24d 151	. . . . . . . . . . . 12 ⊢ (∅𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V)))
24	4, 23	sylbi 217	. . . . . . . . . . 11 ⊢ (∅tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V)))
25	21, 24	biimtrdi 253	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V))))
26	25	com3l 89	. . . . . . . . 9 ⊢ (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V))))
27	26	impcom 407	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
28		brtpos2 8214	. . . . . . . . . . . 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 497	. . . . . . . . . 10 ⊢ (𝑥tpos 𝐹𝑦 → 𝑥 ∈ (◡dom 𝐹 ∪ {∅}))
31		elun 4119	. . . . . . . . . 10 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↔ (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
32	30, 31	sylib 218	. . . . . . . . 9 ⊢ (𝑥tpos 𝐹𝑦 → (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
33	32	adantl 481	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
34	19, 27, 33	mpjaod 860	. . . . . . 7 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
35	34	ex 412	. . . . . 6 ⊢ (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦 → 𝑥 ∈ (V × V)))
36	35	exlimdv 1933	. . . . 5 ⊢ (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦 → 𝑥 ∈ (V × V)))
37	14, 36	biimtrid 242	. . . 4 ⊢ (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹 → 𝑥 ∈ (V × V)))
38	37	ssrdv 3955	. . 3 ⊢ (¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ (V × V))
39		df-rel 5648	. . 3 ⊢ (Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ (V × V))
40	38, 39	sylibr 234	. 2 ⊢ (¬ ∅ ∈ dom 𝐹 → Rel dom tpos 𝐹)
41	12, 40	impbii 209	1 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  ∃wex 1779   ∈ wcel 2109  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  ∅c0 4299  {csn 4592  ∪ cuni 4874   class class class wbr 5110   × cxp 5639  ^◡ccnv 5640  dom cdm 5641  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:  dmtpos  8220  2oppf  49125