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Theorem dmtpos 7896
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 5577 . . . . 5 ¬ ∅ ∈ (V × V)
2 ssel 3946 . . . . 5 (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V)))
31, 2mtoi 202 . . . 4 (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹)
4 df-rel 5550 . . . 4 (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V))
5 reldmtpos 7892 . . . 4 (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
63, 4, 53imtr4i 295 . . 3 (Rel dom 𝐹 → Rel dom tpos 𝐹)
7 relcnv 5955 . . 3 Rel dom 𝐹
86, 7jctir 524 . 2 (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel dom 𝐹))
9 vex 3483 . . . . . 6 𝑧 ∈ V
10 brtpos 7893 . . . . . 6 (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
119, 10mp1i 13 . . . . 5 (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
1211exbidv 1923 . . . 4 (Rel dom 𝐹 → (∃𝑧𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧))
13 opex 5344 . . . . 5 𝑥, 𝑦⟩ ∈ V
1413eldm 5757 . . . 4 (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ∃𝑧𝑥, 𝑦⟩tpos 𝐹𝑧)
15 vex 3483 . . . . . 6 𝑥 ∈ V
16 vex 3483 . . . . . 6 𝑦 ∈ V
1715, 16opelcnv 5740 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom 𝐹)
18 opex 5344 . . . . . 6 𝑦, 𝑥⟩ ∈ V
1918eldm 5757 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧)
2017, 19bitri 278 . . . 4 (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧)
2112, 14, 203bitr4g 317 . . 3 (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
2221eqrelrdv2 5656 . 2 (((Rel dom tpos 𝐹 ∧ Rel dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = dom 𝐹)
238, 22mpancom 687 1 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2115  Vcvv 3480  wss 3919  c0 4276  cop 4556   class class class wbr 5053   × cxp 5541  ccnv 5542  dom cdm 5543  Rel wrel 5548  tpos ctpos 7883
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-fv 6352  df-tpos 7884
This theorem is referenced by:  rntpos  7897  dftpos2  7901  dftpos3  7902  tposfn2  7906
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