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Theorem reldmtpos 8220
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 5300 . . . . 5 ∅ ∈ V
21eldm 5894 . . . 4 (∅ ∈ dom 𝐹 ↔ ∃𝑦𝐹𝑦)
3 brtpos0 8219 . . . . . . 7 (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
43elv 3474 . . . . . 6 (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
5 0nelrel0 5729 . . . . . . 7 (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹)
6 vex 3472 . . . . . . . 8 𝑦 ∈ V
71, 6breldm 5902 . . . . . . 7 (∅tpos 𝐹𝑦 → ∅ ∈ dom tpos 𝐹)
85, 7nsyl3 138 . . . . . 6 (∅tpos 𝐹𝑦 → ¬ Rel dom tpos 𝐹)
94, 8sylbir 234 . . . . 5 (∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
109exlimiv 1925 . . . 4 (∃𝑦𝐹𝑦 → ¬ Rel dom tpos 𝐹)
112, 10sylbi 216 . . 3 (∅ ∈ dom 𝐹 → ¬ Rel dom tpos 𝐹)
1211con2i 139 . 2 (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom 𝐹)
13 vex 3472 . . . . . 6 𝑥 ∈ V
1413eldm 5894 . . . . 5 (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
15 relcnv 6097 . . . . . . . . . . 11 Rel dom 𝐹
16 df-rel 5676 . . . . . . . . . . 11 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
1715, 16mpbi 229 . . . . . . . . . 10 dom 𝐹 ⊆ (V × V)
1817sseli 3973 . . . . . . . . 9 (𝑥dom 𝐹𝑥 ∈ (V × V))
1918a1i 11 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ (V × V)))
20 elsni 4640 . . . . . . . . . . . 12 (𝑥 ∈ {∅} → 𝑥 = ∅)
2120breq1d 5151 . . . . . . . . . . 11 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
221, 6breldm 5902 . . . . . . . . . . . . 13 (∅𝐹𝑦 → ∅ ∈ dom 𝐹)
2322pm2.24d 151 . . . . . . . . . . . 12 (∅𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V)))
244, 23sylbi 216 . . . . . . . . . . 11 (∅tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V)))
2521, 24biimtrdi 252 . . . . . . . . . 10 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V))))
2625com3l 89 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V))))
2726impcom 407 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
28 brtpos2 8218 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦)))
296, 28ax-mp 5 . . . . . . . . . . 11 (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦))
3029simplbi 497 . . . . . . . . . 10 (𝑥tpos 𝐹𝑦𝑥 ∈ (dom 𝐹 ∪ {∅}))
31 elun 4143 . . . . . . . . . 10 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↔ (𝑥dom 𝐹𝑥 ∈ {∅}))
3230, 31sylib 217 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (𝑥dom 𝐹𝑥 ∈ {∅}))
3332adantl 481 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ {∅}))
3419, 27, 33mpjaod 857 . . . . . . 7 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
3534ex 412 . . . . . 6 (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
3635exlimdv 1928 . . . . 5 (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
3714, 36biimtrid 241 . . . 4 (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹𝑥 ∈ (V × V)))
3837ssrdv 3983 . . 3 (¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ (V × V))
39 df-rel 5676 . . 3 (Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ (V × V))
4038, 39sylibr 233 . 2 (¬ ∅ ∈ dom 𝐹 → Rel dom tpos 𝐹)
4112, 40impbii 208 1 (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844  wex 1773  wcel 2098  Vcvv 3468  cun 3941  wss 3943  c0 4317  {csn 4623   cuni 4902   class class class wbr 5141   × cxp 5667  ccnv 5668  dom cdm 5669  Rel wrel 5674  tpos ctpos 8211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545  df-tpos 8212
This theorem is referenced by:  dmtpos  8224
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