MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reldmtpos Structured version   Visualization version   GIF version

Theorem reldmtpos 7405
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.
StepHypRef Expression
1 0ex 4823 . . . . 5 ∅ ∈ V
21eldm 5353 . . . 4 (∅ ∈ dom 𝐹 ↔ ∃𝑦𝐹𝑦)
3 vex 3234 . . . . . . 7 𝑦 ∈ V
4 brtpos0 7404 . . . . . . 7 (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
53, 4ax-mp 5 . . . . . 6 (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
6 0nelxp 5177 . . . . . . . 8 ¬ ∅ ∈ (V × V)
7 df-rel 5150 . . . . . . . . 9 (Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ (V × V))
8 ssel 3630 . . . . . . . . 9 (dom tpos 𝐹 ⊆ (V × V) → (∅ ∈ dom tpos 𝐹 → ∅ ∈ (V × V)))
97, 8sylbi 207 . . . . . . . 8 (Rel dom tpos 𝐹 → (∅ ∈ dom tpos 𝐹 → ∅ ∈ (V × V)))
106, 9mtoi 190 . . . . . . 7 (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹)
111, 3breldm 5361 . . . . . . 7 (∅tpos 𝐹𝑦 → ∅ ∈ dom tpos 𝐹)
1210, 11nsyl3 133 . . . . . 6 (∅tpos 𝐹𝑦 → ¬ Rel dom tpos 𝐹)
135, 12sylbir 225 . . . . 5 (∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
1413exlimiv 1898 . . . 4 (∃𝑦𝐹𝑦 → ¬ Rel dom tpos 𝐹)
152, 14sylbi 207 . . 3 (∅ ∈ dom 𝐹 → ¬ Rel dom tpos 𝐹)
1615con2i 134 . 2 (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom 𝐹)
17 vex 3234 . . . . . 6 𝑥 ∈ V
1817eldm 5353 . . . . 5 (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
19 relcnv 5538 . . . . . . . . . . 11 Rel dom 𝐹
20 df-rel 5150 . . . . . . . . . . 11 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
2119, 20mpbi 220 . . . . . . . . . 10 dom 𝐹 ⊆ (V × V)
2221sseli 3632 . . . . . . . . 9 (𝑥dom 𝐹𝑥 ∈ (V × V))
2322a1i 11 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ (V × V)))
24 elsni 4227 . . . . . . . . . . . 12 (𝑥 ∈ {∅} → 𝑥 = ∅)
2524breq1d 4695 . . . . . . . . . . 11 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
261, 3breldm 5361 . . . . . . . . . . . . 13 (∅𝐹𝑦 → ∅ ∈ dom 𝐹)
2726pm2.24d 147 . . . . . . . . . . . 12 (∅𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V)))
285, 27sylbi 207 . . . . . . . . . . 11 (∅tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V)))
2925, 28syl6bi 243 . . . . . . . . . 10 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V))))
3029com3l 89 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V))))
3130impcom 445 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
32 brtpos2 7403 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦)))
333, 32ax-mp 5 . . . . . . . . . . 11 (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦))
3433simplbi 475 . . . . . . . . . 10 (𝑥tpos 𝐹𝑦𝑥 ∈ (dom 𝐹 ∪ {∅}))
35 elun 3786 . . . . . . . . . 10 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↔ (𝑥dom 𝐹𝑥 ∈ {∅}))
3634, 35sylib 208 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (𝑥dom 𝐹𝑥 ∈ {∅}))
3736adantl 481 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ {∅}))
3823, 31, 37mpjaod 395 . . . . . . 7 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
3938ex 449 . . . . . 6 (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
4039exlimdv 1901 . . . . 5 (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
4118, 40syl5bi 232 . . . 4 (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹𝑥 ∈ (V × V)))
4241ssrdv 3642 . . 3 (¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ (V × V))
4342, 7sylibr 224 . 2 (¬ ∅ ∈ dom 𝐹 → Rel dom tpos 𝐹)
4416, 43impbii 199 1 (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  wex 1744  wcel 2030  Vcvv 3231  cun 3605  wss 3607  c0 3948  {csn 4210   cuni 4468   class class class wbr 4685   × cxp 5141  ccnv 5142  dom cdm 5143  Rel wrel 5148  tpos ctpos 7396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-tpos 7397
This theorem is referenced by:  dmtpos  7409
  Copyright terms: Public domain W3C validator