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Theorem dmtpos 7646
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 5389 . . . . 5 ¬ ∅ ∈ (V × V)
2 ssel 3814 . . . . 5 (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V)))
31, 2mtoi 191 . . . 4 (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹)
4 df-rel 5362 . . . 4 (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V))
5 reldmtpos 7642 . . . 4 (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
63, 4, 53imtr4i 284 . . 3 (Rel dom 𝐹 → Rel dom tpos 𝐹)
7 relcnv 5757 . . 3 Rel dom 𝐹
86, 7jctir 516 . 2 (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel dom 𝐹))
9 vex 3400 . . . . . 6 𝑧 ∈ V
10 brtpos 7643 . . . . . 6 (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
119, 10mp1i 13 . . . . 5 (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
1211exbidv 1964 . . . 4 (Rel dom 𝐹 → (∃𝑧𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧))
13 opex 5164 . . . . 5 𝑥, 𝑦⟩ ∈ V
1413eldm 5566 . . . 4 (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ∃𝑧𝑥, 𝑦⟩tpos 𝐹𝑧)
15 vex 3400 . . . . . 6 𝑥 ∈ V
16 vex 3400 . . . . . 6 𝑦 ∈ V
1715, 16opelcnv 5549 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom 𝐹)
18 opex 5164 . . . . . 6 𝑦, 𝑥⟩ ∈ V
1918eldm 5566 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧)
2017, 19bitri 267 . . . 4 (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑥𝐹𝑧)
2112, 14, 203bitr4g 306 . . 3 (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
2221eqrelrdv2 5466 . 2 (((Rel dom tpos 𝐹 ∧ Rel dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = dom 𝐹)
238, 22mpancom 678 1 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1601  wex 1823  wcel 2106  Vcvv 3397  wss 3791  c0 4140  cop 4403   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 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  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 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  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:  rntpos  7647  dftpos2  7651  dftpos3  7652  tposfn2  7656
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