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Theorem reldmtpos 7642
 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 5026 . . . . 5 ∅ ∈ V
21eldm 5566 . . . 4 (∅ ∈ dom 𝐹 ↔ ∃𝑦𝐹𝑦)
3 brtpos0 7641 . . . . . . 7 (𝑦 ∈ V → (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦))
43elv 3402 . . . . . 6 (∅tpos 𝐹𝑦 ↔ ∅𝐹𝑦)
5 0nelxp 5389 . . . . . . . 8 ¬ ∅ ∈ (V × V)
6 df-rel 5362 . . . . . . . . 9 (Rel dom tpos 𝐹 ↔ dom tpos 𝐹 ⊆ (V × V))
7 ssel 3815 . . . . . . . . 9 (dom tpos 𝐹 ⊆ (V × V) → (∅ ∈ dom tpos 𝐹 → ∅ ∈ (V × V)))
86, 7sylbi 209 . . . . . . . 8 (Rel dom tpos 𝐹 → (∅ ∈ dom tpos 𝐹 → ∅ ∈ (V × V)))
95, 8mtoi 191 . . . . . . 7 (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom tpos 𝐹)
10 vex 3401 . . . . . . . 8 𝑦 ∈ V
111, 10breldm 5574 . . . . . . 7 (∅tpos 𝐹𝑦 → ∅ ∈ dom tpos 𝐹)
129, 11nsyl3 136 . . . . . 6 (∅tpos 𝐹𝑦 → ¬ Rel dom tpos 𝐹)
134, 12sylbir 227 . . . . 5 (∅𝐹𝑦 → ¬ Rel dom tpos 𝐹)
1413exlimiv 1973 . . . 4 (∃𝑦𝐹𝑦 → ¬ Rel dom tpos 𝐹)
152, 14sylbi 209 . . 3 (∅ ∈ dom 𝐹 → ¬ Rel dom tpos 𝐹)
1615con2i 137 . 2 (Rel dom tpos 𝐹 → ¬ ∅ ∈ dom 𝐹)
17 vex 3401 . . . . . 6 𝑥 ∈ V
1817eldm 5566 . . . . 5 (𝑥 ∈ dom tpos 𝐹 ↔ ∃𝑦 𝑥tpos 𝐹𝑦)
19 relcnv 5757 . . . . . . . . . . 11 Rel dom 𝐹
20 df-rel 5362 . . . . . . . . . . 11 (Rel dom 𝐹dom 𝐹 ⊆ (V × V))
2119, 20mpbi 222 . . . . . . . . . 10 dom 𝐹 ⊆ (V × V)
2221sseli 3817 . . . . . . . . 9 (𝑥dom 𝐹𝑥 ∈ (V × V))
2322a1i 11 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ (V × V)))
24 elsni 4415 . . . . . . . . . . . 12 (𝑥 ∈ {∅} → 𝑥 = ∅)
2524breq1d 4896 . . . . . . . . . . 11 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 ↔ ∅tpos 𝐹𝑦))
261, 10breldm 5574 . . . . . . . . . . . . 13 (∅𝐹𝑦 → ∅ ∈ dom 𝐹)
2726pm2.24d 149 . . . . . . . . . . . 12 (∅𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V)))
284, 27sylbi 209 . . . . . . . . . . 11 (∅tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V)))
2925, 28syl6bi 245 . . . . . . . . . 10 (𝑥 ∈ {∅} → (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹𝑥 ∈ (V × V))))
3029com3l 89 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V))))
3130impcom 398 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥 ∈ {∅} → 𝑥 ∈ (V × V)))
32 brtpos2 7640 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦)))
3332elv 3402 . . . . . . . . . . 11 (𝑥tpos 𝐹𝑦 ↔ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝑥}𝐹𝑦))
3433simplbi 493 . . . . . . . . . 10 (𝑥tpos 𝐹𝑦𝑥 ∈ (dom 𝐹 ∪ {∅}))
35 elun 3976 . . . . . . . . . 10 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↔ (𝑥dom 𝐹𝑥 ∈ {∅}))
3634, 35sylib 210 . . . . . . . . 9 (𝑥tpos 𝐹𝑦 → (𝑥dom 𝐹𝑥 ∈ {∅}))
3736adantl 475 . . . . . . . 8 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → (𝑥dom 𝐹𝑥 ∈ {∅}))
3823, 31, 37mpjaod 849 . . . . . . 7 ((¬ ∅ ∈ dom 𝐹𝑥tpos 𝐹𝑦) → 𝑥 ∈ (V × V))
3938ex 403 . . . . . 6 (¬ ∅ ∈ dom 𝐹 → (𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
4039exlimdv 1976 . . . . 5 (¬ ∅ ∈ dom 𝐹 → (∃𝑦 𝑥tpos 𝐹𝑦𝑥 ∈ (V × V)))
4118, 40syl5bi 234 . . . 4 (¬ ∅ ∈ dom 𝐹 → (𝑥 ∈ dom tpos 𝐹𝑥 ∈ (V × V)))
4241ssrdv 3827 . . 3 (¬ ∅ ∈ dom 𝐹 → dom tpos 𝐹 ⊆ (V × V))
4342, 6sylibr 226 . 2 (¬ ∅ ∈ dom 𝐹 → Rel dom tpos 𝐹)
4416, 43impbii 201 1 (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836  ∃wex 1823   ∈ wcel 2107  Vcvv 3398   ∪ cun 3790   ⊆ wss 3792  ∅c0 4141  {csn 4398  ∪ cuni 4671   class class class wbr 4886   × cxp 5353  ◡ccnv 5354  dom cdm 5355  Rel wrel 5360  tpos ctpos 7633 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143  df-tpos 7634 This theorem is referenced by:  dmtpos  7646
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