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Theorem dmtpos 6107
Description: The domain of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
dmtpos (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)

Proof of Theorem dmtpos
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0nelxp 4527 . . . . 5 ¬ ∅ ∈ (V × V)
2 ssel 3057 . . . . 5 (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V)))
31, 2mtoi 636 . . . 4 (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹)
4 df-rel 4506 . . . 4 (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V))
5 reldmtpos 6104 . . . 4 (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
63, 4, 53imtr4i 200 . . 3 (Rel dom 𝐹 → Rel dom tpos 𝐹)
7 relcnv 4875 . . 3 Rel dom 𝐹
86, 7jctir 309 . 2 (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel dom 𝐹))
9 vex 2660 . . . . . . 7 𝑥 ∈ V
10 vex 2660 . . . . . . 7 𝑦 ∈ V
11 vex 2660 . . . . . . 7 𝑧 ∈ V
12 brtposg 6105 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
139, 10, 11, 12mp3an 1298 . . . . . 6 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧)
1413a1i 9 . . . . 5 (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
1514exbidv 1779 . . . 4 (Rel dom 𝐹 → (∃𝑧𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧))
169, 10opex 4111 . . . . 5 𝑥, 𝑦⟩ ∈ V
1716eldm 4696 . . . 4 (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ∃𝑧𝑥, 𝑦⟩tpos 𝐹𝑧)
189, 10opelcnv 4681 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom 𝐹)
1910, 9opex 4111 . . . . . 6 𝑦, 𝑥⟩ ∈ V
2019eldm 4696 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧)
2118, 20bitri 183 . . . 4 (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧)
2215, 17, 213bitr4g 222 . . 3 (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
2322eqrelrdv2 4598 . 2 (((Rel dom tpos 𝐹 ∧ Rel dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = dom 𝐹)
248, 23mpancom 416 1 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104   = wceq 1314  wex 1451  wcel 1463  Vcvv 2657  wss 3037  c0 3329  cop 3496   class class class wbr 3895   × cxp 4497  ccnv 4498  dom cdm 4499  Rel wrel 4504  tpos ctpos 6095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-fv 5089  df-tpos 6096
This theorem is referenced by:  rntpos  6108  dftpos2  6112  dftpos3  6113  tposfn2  6117
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