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Theorem reldmtpos 7889
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 5202 . . . . 5 ∅ ∈ V
21eldm 5762 . . . 4 (∅ ∈ dom 𝐹 ↔ ∃𝑦𝐹𝑦)
3 brtpos0 7888 . . . . . . 7 (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
43elv 3497 . . . . . 6 (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
5 0nelrel0 5605 . . . . . . 7 (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹)
6 vex 3495 . . . . . . . 8 𝑦 ∈ V
71, 6breldm 5770 . . . . . . 7 (∅tpos 𝐹𝑦 → ∅ ∈ dom tpos 𝐹)
85, 7nsyl3 140 . . . . . 6 (∅tpos 𝐹𝑦 → ¬ Rel dom tpos 𝐹)
94, 8sylbir 236 . . . . 5 (∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
109exlimiv 1922 . . . 4 (∃𝑦𝐹𝑦 → ¬ Rel dom tpos 𝐹)
112, 10sylbi 218 . . 3 (∅ ∈ dom 𝐹 → ¬ Rel dom tpos 𝐹)
1211con2i 141 . 2 (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom 𝐹)
13 vex 3495 . . . . . 6 𝑥 ∈ V
1413eldm 5762 . . . . 5 (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
15 relcnv 5960 . . . . . . . . . . 11 Rel dom 𝐹
16 df-rel 5555 . . . . . . . . . . 11 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
1715, 16mpbi 231 . . . . . . . . . 10 dom 𝐹 ⊆ (V × V)
1817sseli 3960 . . . . . . . . 9 (𝑥dom 𝐹𝑥 ∈ (V × V))
1918a1i 11 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ (V × V)))
20 elsni 4574 . . . . . . . . . . . 12 (𝑥 ∈ {∅} → 𝑥 = ∅)
2120breq1d 5067 . . . . . . . . . . 11 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
221, 6breldm 5770 . . . . . . . . . . . . 13 (∅𝐹𝑦 → ∅ ∈ dom 𝐹)
2322pm2.24d 154 . . . . . . . . . . . 12 (∅𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V)))
244, 23sylbi 218 . . . . . . . . . . 11 (∅tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V)))
2521, 24syl6bi 254 . . . . . . . . . 10 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V))))
2625com3l 89 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V))))
2726impcom 408 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
28 brtpos2 7887 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦)))
296, 28ax-mp 5 . . . . . . . . . . 11 (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦))
3029simplbi 498 . . . . . . . . . 10 (𝑥tpos 𝐹𝑦𝑥 ∈ (dom 𝐹 ∪ {∅}))
31 elun 4122 . . . . . . . . . 10 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↔ (𝑥dom 𝐹𝑥 ∈ {∅}))
3230, 31sylib 219 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (𝑥dom 𝐹𝑥 ∈ {∅}))
3332adantl 482 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ {∅}))
3419, 27, 33mpjaod 854 . . . . . . 7 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
3534ex 413 . . . . . 6 (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
3635exlimdv 1925 . . . . 5 (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
3714, 36syl5bi 243 . . . 4 (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹𝑥 ∈ (V × V)))
3837ssrdv 3970 . . 3 (¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ (V × V))
39 df-rel 5555 . . 3 (Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ (V × V))
4038, 39sylibr 235 . 2 (¬ ∅ ∈ dom 𝐹 → Rel dom tpos 𝐹)
4112, 40impbii 210 1 (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  wex 1771  wcel 2105  Vcvv 3492  cun 3931  wss 3933  c0 4288  {csn 4557   cuni 4830   class class class wbr 5057   × cxp 5546  ccnv 5547  dom cdm 5548  Rel wrel 5553  tpos ctpos 7880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-tpos 7881
This theorem is referenced by:  dmtpos  7893
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