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Theorem reldmtpos 7357
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 4788 . . . . 5 ∅ ∈ V
21eldm 5319 . . . 4 (∅ ∈ dom 𝐹 ↔ ∃𝑦𝐹𝑦)
3 vex 3201 . . . . . . 7 𝑦 ∈ V
4 brtpos0 7356 . . . . . . 7 (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
53, 4ax-mp 5 . . . . . 6 (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
6 0nelxp 5141 . . . . . . . 8 ¬ ∅ ∈ (V × V)
7 df-rel 5119 . . . . . . . . 9 (Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ (V × V))
8 ssel 3595 . . . . . . . . 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 5327 . . . . . . 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 1857 . . . 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 3201 . . . . . 6 𝑥 ∈ V
1817eldm 5319 . . . . 5 (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
19 relcnv 5501 . . . . . . . . . . 11 Rel dom 𝐹
20 df-rel 5119 . . . . . . . . . . 11 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
2119, 20mpbi 220 . . . . . . . . . 10 dom 𝐹 ⊆ (V × V)
2221sseli 3597 . . . . . . . . 9 (𝑥dom 𝐹𝑥 ∈ (V × V))
2322a1i 11 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ (V × V)))
24 elsni 4192 . . . . . . . . . . . 12 (𝑥 ∈ {∅} → 𝑥 = ∅)
2524breq1d 4661 . . . . . . . . . . 11 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
261, 3breldm 5327 . . . . . . . . . . . . 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 446 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
32 brtpos2 7355 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦)))
333, 32ax-mp 5 . . . . . . . . . . 11 (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦))
3433simplbi 476 . . . . . . . . . 10 (𝑥tpos 𝐹𝑦𝑥 ∈ (dom 𝐹 ∪ {∅}))
35 elun 3751 . . . . . . . . . 10 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↔ (𝑥dom 𝐹𝑥 ∈ {∅}))
3634, 35sylib 208 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (𝑥dom 𝐹𝑥 ∈ {∅}))
3736adantl 482 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ {∅}))
3823, 31, 37mpjaod 396 . . . . . . 7 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
3938ex 450 . . . . . 6 (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
4039exlimdv 1860 . . . . 5 (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
4118, 40syl5bi 232 . . . 4 (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹𝑥 ∈ (V × V)))
4241ssrdv 3607 . . 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 383  wa 384  wex 1703  wcel 1989  Vcvv 3198  cun 3570  wss 3572  c0 3913  {csn 4175   cuni 4434   class class class wbr 4651   × cxp 5110  ccnv 5111  dom cdm 5112  Rel wrel 5117  tpos ctpos 7348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894  df-tpos 7349
This theorem is referenced by:  dmtpos  7361
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