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Theorem reldmtpos 7887
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 5187 . . . . 5 ∅ ∈ V
21eldm 5746 . . . 4 (∅ ∈ dom 𝐹 ↔ ∃𝑦𝐹𝑦)
3 brtpos0 7886 . . . . . . 7 (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
43elv 3474 . . . . . 6 (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
5 0nelrel0 5589 . . . . . . 7 (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹)
6 vex 3472 . . . . . . . 8 𝑦 ∈ V
71, 6breldm 5754 . . . . . . 7 (∅tpos 𝐹𝑦 → ∅ ∈ dom tpos 𝐹)
85, 7nsyl3 140 . . . . . 6 (∅tpos 𝐹𝑦 → ¬ Rel dom tpos 𝐹)
94, 8sylbir 238 . . . . 5 (∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
109exlimiv 1931 . . . 4 (∃𝑦𝐹𝑦 → ¬ Rel dom tpos 𝐹)
112, 10sylbi 220 . . 3 (∅ ∈ dom 𝐹 → ¬ Rel dom tpos 𝐹)
1211con2i 141 . 2 (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom 𝐹)
13 vex 3472 . . . . . 6 𝑥 ∈ V
1413eldm 5746 . . . . 5 (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
15 relcnv 5945 . . . . . . . . . . 11 Rel dom 𝐹
16 df-rel 5539 . . . . . . . . . . 11 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
1715, 16mpbi 233 . . . . . . . . . 10 dom 𝐹 ⊆ (V × V)
1817sseli 3938 . . . . . . . . 9 (𝑥dom 𝐹𝑥 ∈ (V × V))
1918a1i 11 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ (V × V)))
20 elsni 4556 . . . . . . . . . . . 12 (𝑥 ∈ {∅} → 𝑥 = ∅)
2120breq1d 5052 . . . . . . . . . . 11 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
221, 6breldm 5754 . . . . . . . . . . . . 13 (∅𝐹𝑦 → ∅ ∈ dom 𝐹)
2322pm2.24d 154 . . . . . . . . . . . 12 (∅𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V)))
244, 23sylbi 220 . . . . . . . . . . 11 (∅tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V)))
2521, 24syl6bi 256 . . . . . . . . . 10 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V))))
2625com3l 89 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V))))
2726impcom 411 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
28 brtpos2 7885 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦)))
296, 28ax-mp 5 . . . . . . . . . . 11 (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦))
3029simplbi 501 . . . . . . . . . 10 (𝑥tpos 𝐹𝑦𝑥 ∈ (dom 𝐹 ∪ {∅}))
31 elun 4100 . . . . . . . . . 10 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↔ (𝑥dom 𝐹𝑥 ∈ {∅}))
3230, 31sylib 221 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (𝑥dom 𝐹𝑥 ∈ {∅}))
3332adantl 485 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ {∅}))
3419, 27, 33mpjaod 857 . . . . . . 7 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
3534ex 416 . . . . . 6 (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
3635exlimdv 1934 . . . . 5 (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
3714, 36syl5bi 245 . . . 4 (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹𝑥 ∈ (V × V)))
3837ssrdv 3948 . . 3 (¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ (V × V))
39 df-rel 5539 . . 3 (Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ (V × V))
4038, 39sylibr 237 . 2 (¬ ∅ ∈ dom 𝐹 → Rel dom tpos 𝐹)
4112, 40impbii 212 1 (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  wex 1781  wcel 2114  Vcvv 3469  cun 3906  wss 3908  c0 4265  {csn 4539   cuni 4813   class class class wbr 5042   × cxp 5530  ccnv 5531  dom cdm 5532  Rel wrel 5537  tpos ctpos 7878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342  df-tpos 7879
This theorem is referenced by:  dmtpos  7891
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