Theorem reldmtpos 8095

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 5244	. . . . 5 ⊢ ∅ ∈ V
2	1	eldm 5827	. . . 4 ⊢ (∅ ∈ dom 𝐹 ↔ ∃𝑦∅𝐹𝑦)
3		brtpos0 8094	. . . . . . 7 ⊢ (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
4	3	elv 3447	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
5		0nelrel0 5663	. . . . . . 7 ⊢ (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹)
6		vex 3445	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	1, 6	breldm 5835	. . . . . . 7 ⊢ (∅tpos 𝐹𝑦 → ∅ ∈ dom tpos 𝐹)
8	5, 7	nsyl3 138	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 → ¬ Rel dom tpos 𝐹)
9	4, 8	sylbir 234	. . . . 5 ⊢ (∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
10	9	exlimiv 1932	. . . 4 ⊢ (∃𝑦∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
11	2, 10	sylbi 216	. . 3 ⊢ (∅ ∈ dom 𝐹 → ¬ Rel dom tpos 𝐹)
12	11	con2i 139	. 2 ⊢ (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom 𝐹)
13		vex 3445	. . . . . 6 ⊢ 𝑥 ∈ V
14	13	eldm 5827	. . . . 5 ⊢ (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
15		relcnv 6027	. . . . . . . . . . 11 ⊢ Rel ◡dom 𝐹
16		df-rel 5612	. . . . . . . . . . 11 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
17	15, 16	mpbi 229	. . . . . . . . . 10 ⊢ ◡dom 𝐹 ⊆ (V × V)
18	17	sseli 3926	. . . . . . . . 9 ⊢ (𝑥 ∈ ◡dom 𝐹 → 𝑥 ∈ (V × V))
19	18	a1i 11	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ ◡dom 𝐹 → 𝑥 ∈ (V × V)))
20		elsni 4586	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
21	20	breq1d 5095	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
22	1, 6	breldm 5835	. . . . . . . . . . . . 13 ⊢ (∅𝐹𝑦 → ∅ ∈ dom 𝐹)
23	22	pm2.24d 151	. . . . . . . . . . . 12 ⊢ (∅𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V)))
24	4, 23	sylbi 216	. . . . . . . . . . 11 ⊢ (∅tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V)))
25	21, 24	syl6bi 252	. . . . . . . . . 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 408	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
28		brtpos2 8093	. . . . . . . . . . . 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 498	. . . . . . . . . 10 ⊢ (𝑥tpos 𝐹𝑦 → 𝑥 ∈ (◡dom 𝐹 ∪ {∅}))
31		elun 4093	. . . . . . . . . 10 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↔ (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
32	30, 31	sylib 217	. . . . . . . . 9 ⊢ (𝑥tpos 𝐹𝑦 → (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
33	32	adantl 482	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
34	19, 27, 33	mpjaod 857	. . . . . . 7 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
35	34	ex 413	. . . . . 6 ⊢ (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦 → 𝑥 ∈ (V × V)))
36	35	exlimdv 1935	. . . . 5 ⊢ (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦 → 𝑥 ∈ (V × V)))
37	14, 36	biimtrid 241	. . . 4 ⊢ (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹 → 𝑥 ∈ (V × V)))
38	37	ssrdv 3936	. . 3 ⊢ (¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ (V × V))
39		df-rel 5612	. . 3 ⊢ (Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ (V × V))
40	38, 39	sylibr 233	. 2 ⊢ (¬ ∅ ∈ dom 𝐹 → Rel dom tpos 𝐹)
41	12, 40	impbii 208	1 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844  ∃wex 1780   ∈ wcel 2105  Vcvv 3441   ∪ cun 3894   ⊆ wss 3896  ∅c0 4266  {csn 4569  ∪ cuni 4848   class class class wbr 5085   × cxp 5603  ^◡ccnv 5604  dom cdm 5605  Rel wrel 5610  tpos ctpos 8086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-fv 6471  df-tpos 8087

This theorem is referenced by:  dmtpos  8099