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Theorem dmtpos 8173
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 5671 . . . . 5 ¬ ∅ ∈ (V × V)
2 ssel 3941 . . . . 5 (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V)))
31, 2mtoi 198 . . . 4 (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹)
4 df-rel 5644 . . . 4 (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V))
5 reldmtpos 8169 . . . 4 (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
63, 4, 53imtr4i 292 . . 3 (Rel dom 𝐹 → Rel dom tpos 𝐹)
7 relcnv 6060 . . 3 Rel dom 𝐹
86, 7jctir 522 . 2 (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel dom 𝐹))
9 vex 3451 . . . . . 6 𝑧 ∈ V
10 brtpos 8170 . . . . . 6 (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
119, 10mp1i 13 . . . . 5 (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
1211exbidv 1925 . . . 4 (Rel dom 𝐹 → (∃𝑧𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧))
13 opex 5425 . . . . 5 𝑥, 𝑦⟩ ∈ V
1413eldm 5860 . . . 4 (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ∃𝑧𝑥, 𝑦⟩tpos 𝐹𝑧)
15 vex 3451 . . . . . 6 𝑥 ∈ V
16 vex 3451 . . . . . 6 𝑦 ∈ V
1715, 16opelcnv 5841 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom 𝐹)
18 opex 5425 . . . . . 6 𝑦, 𝑥⟩ ∈ V
1918eldm 5860 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧)
2017, 19bitri 275 . . . 4 (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧)
2112, 14, 203bitr4g 314 . . 3 (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
2221eqrelrdv2 5755 . 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 205  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3447  wss 3914  c0 4286  cop 4596   class class class wbr 5109   × cxp 5635  ccnv 5636  dom cdm 5637  Rel wrel 5642  tpos ctpos 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-tpos 8161
This theorem is referenced by:  rntpos  8174  dftpos2  8178  dftpos3  8179  tposfn2  8183
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