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Theorem reldmtpos 8241
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 5304 . . . . 5 ∅ ∈ V
21eldm 5899 . . . 4 (∅ ∈ dom 𝐹 ↔ ∃𝑦𝐹𝑦)
3 brtpos0 8240 . . . . . . 7 (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
43elv 3468 . . . . . 6 (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
5 0nelrel0 5734 . . . . . . 7 (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹)
6 vex 3466 . . . . . . . 8 𝑦 ∈ V
71, 6breldm 5907 . . . . . . 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 1926 . . . 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 3466 . . . . . 6 𝑥 ∈ V
1413eldm 5899 . . . . 5 (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
15 relcnv 6106 . . . . . . . . . . 11 Rel dom 𝐹
16 df-rel 5681 . . . . . . . . . . 11 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
1715, 16mpbi 229 . . . . . . . . . 10 dom 𝐹 ⊆ (V × V)
1817sseli 3974 . . . . . . . . 9 (𝑥dom 𝐹𝑥 ∈ (V × V))
1918a1i 11 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ (V × V)))
20 elsni 4640 . . . . . . . . . . . 12 (𝑥 ∈ {∅} → 𝑥 = ∅)
2120breq1d 5155 . . . . . . . . . . 11 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
221, 6breldm 5907 . . . . . . . . . . . . 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 406 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
28 brtpos2 8239 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦)))
296, 28ax-mp 5 . . . . . . . . . . 11 (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦))
3029simplbi 496 . . . . . . . . . 10 (𝑥tpos 𝐹𝑦𝑥 ∈ (dom 𝐹 ∪ {∅}))
31 elun 4145 . . . . . . . . . 10 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↔ (𝑥dom 𝐹𝑥 ∈ {∅}))
3230, 31sylib 217 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (𝑥dom 𝐹𝑥 ∈ {∅}))
3332adantl 480 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ {∅}))
3419, 27, 33mpjaod 858 . . . . . . 7 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
3534ex 411 . . . . . 6 (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
3635exlimdv 1929 . . . . 5 (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
3714, 36biimtrid 241 . . . 4 (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹𝑥 ∈ (V × V)))
3837ssrdv 3984 . . 3 (¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ (V × V))
39 df-rel 5681 . . 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 394  wo 845  wex 1774  wcel 2099  Vcvv 3462  cun 3944  wss 3946  c0 4322  {csn 4623   cuni 4905   class class class wbr 5145   × cxp 5672  ccnv 5673  dom cdm 5674  Rel wrel 5679  tpos ctpos 8232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-fv 6554  df-tpos 8233
This theorem is referenced by:  dmtpos  8245
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