Theorem reldmtpos 6462

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 4221	. . . . 5 ⊢ ∅ ∈ V
2	1	eldm 4934	. . . 4 ⊢ (∅ ∈ dom 𝐹 ↔ ∃𝑦∅𝐹𝑦)
3		vex 2806	. . . . . . 7 ⊢ 𝑦 ∈ V
4		brtpos0 6461	. . . . . . 7 ⊢ (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
5	3, 4	ax-mp 5	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
6		0nelxp 4759	. . . . . . . 8 ⊢ ¬ ∅ ∈ (V × V)
7		df-rel 4738	. . . . . . . . 9 ⊢ (Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ (V × V))
8		ssel 3222	. . . . . . . . 9 ⊢ (dom tpos 𝐹 ⊆ (V × V) → (∅ ∈ dom tpos 𝐹 → ∅ ∈ (V × V)))
9	7, 8	sylbi 121	. . . . . . . 8 ⊢ (Rel dom tpos 𝐹 → (∅ ∈ dom tpos 𝐹 → ∅ ∈ (V × V)))
10	6, 9	mtoi 670	. . . . . . 7 ⊢ (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹)
11	1, 3	breldm 4941	. . . . . . 7 ⊢ (∅tpos 𝐹𝑦 → ∅ ∈ dom tpos 𝐹)
12	10, 11	nsyl3 631	. . . . . 6 ⊢ (∅tpos 𝐹𝑦 → ¬ Rel dom tpos 𝐹)
13	5, 12	sylbir 135	. . . . 5 ⊢ (∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
14	13	exlimiv 1647	. . . 4 ⊢ (∃𝑦∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
15	2, 14	sylbi 121	. . 3 ⊢ (∅ ∈ dom 𝐹 → ¬ Rel dom tpos 𝐹)
16	15	con2i 632	. 2 ⊢ (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom 𝐹)
17		vex 2806	. . . . . 6 ⊢ 𝑥 ∈ V
18	17	eldm 4934	. . . . 5 ⊢ (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
19		relcnv 5121	. . . . . . . . . . 11 ⊢ Rel ◡dom 𝐹
20		df-rel 4738	. . . . . . . . . . 11 ⊢ (Rel ◡dom 𝐹 ↔ ◡dom 𝐹 ⊆ (V × V))
21	19, 20	mpbi 145	. . . . . . . . . 10 ⊢ ◡dom 𝐹 ⊆ (V × V)
22	21	sseli 3224	. . . . . . . . 9 ⊢ (𝑥 ∈ ◡dom 𝐹 → 𝑥 ∈ (V × V))
23	22	a1i 9	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ ◡dom 𝐹 → 𝑥 ∈ (V × V)))
24		elsni 3691	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
25	24	breq1d 4103	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
26	1, 3	breldm 4941	. . . . . . . . . . . . 13 ⊢ (∅𝐹𝑦 → ∅ ∈ dom 𝐹)
27	26	pm2.24d 627	. . . . . . . . . . . 12 ⊢ (∅𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V)))
28	5, 27	sylbi 121	. . . . . . . . . . 11 ⊢ (∅tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V)))
29	25, 28	biimtrdi 163	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → 𝑥 ∈ (V × V))))
30	29	com3l 81	. . . . . . . . 9 ⊢ (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V))))
31	30	impcom 125	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
32		brtpos2 6460	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑥}𝐹𝑦)))
33	3, 32	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑥}𝐹𝑦))
34	33	simplbi 274	. . . . . . . . . 10 ⊢ (𝑥tpos 𝐹𝑦 → 𝑥 ∈ (◡dom 𝐹 ∪ {∅}))
35		elun 3350	. . . . . . . . . 10 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↔ (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
36	34, 35	sylib 122	. . . . . . . . 9 ⊢ (𝑥tpos 𝐹𝑦 → (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
37	36	adantl 277	. . . . . . . 8 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → (𝑥 ∈ ◡dom 𝐹 ∨ 𝑥 ∈ {∅}))
38	23, 31, 37	mpjaod 726	. . . . . . 7 ⊢ ((¬ ∅ ∈ dom 𝐹 ∧ 𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
39	38	ex 115	. . . . . 6 ⊢ (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦 → 𝑥 ∈ (V × V)))
40	39	exlimdv 1867	. . . . 5 ⊢ (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦 → 𝑥 ∈ (V × V)))
41	18, 40	biimtrid 152	. . . 4 ⊢ (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹 → 𝑥 ∈ (V × V)))
42	41	ssrdv 3234	. . 3 ⊢ (¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ (V × V))
43	42, 7	sylibr 134	. 2 ⊢ (¬ ∅ ∈ dom 𝐹 → Rel dom tpos 𝐹)
44	16, 43	impbii 126	1 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  ∃wex 1541   ∈ wcel 2202  Vcvv 2803   ∪ cun 3199   ⊆ wss 3201  ∅c0 3496  {csn 3673  ∪ cuni 3898   class class class wbr 4093   × cxp 4729  ^◡ccnv 4730  dom cdm 4731  Rel wrel 4736  tpos ctpos 6453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-tpos 6454

This theorem is referenced by:  dmtpos  6465