Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfcnv2 Structured version   Visualization version   GIF version

Theorem dfcnv2 32693
Description: Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.)
Assertion
Ref Expression
dfcnv2 (ran 𝑅𝐴𝑅 = 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfcnv2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 6125 . 2 Rel 𝑅
2 relxp 5707 . . . 4 Rel ({𝑥} × (𝑅 “ {𝑥}))
32rgenw 3063 . . 3 𝑥𝐴 Rel ({𝑥} × (𝑅 “ {𝑥}))
4 reliun 5829 . . 3 (Rel 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})) ↔ ∀𝑥𝐴 Rel ({𝑥} × (𝑅 “ {𝑥})))
53, 4mpbir 231 . 2 Rel 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥}))
6 vex 3482 . . . . . . . . 9 𝑧 ∈ V
7 vex 3482 . . . . . . . . 9 𝑦 ∈ V
86, 7opeldm 5921 . . . . . . . 8 (⟨𝑧, 𝑦⟩ ∈ 𝑅𝑧 ∈ dom 𝑅)
9 df-rn 5700 . . . . . . . 8 ran 𝑅 = dom 𝑅
108, 9eleqtrrdi 2850 . . . . . . 7 (⟨𝑧, 𝑦⟩ ∈ 𝑅𝑧 ∈ ran 𝑅)
11 ssel2 3990 . . . . . . 7 ((ran 𝑅𝐴𝑧 ∈ ran 𝑅) → 𝑧𝐴)
1210, 11sylan2 593 . . . . . 6 ((ran 𝑅𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑅) → 𝑧𝐴)
1312ex 412 . . . . 5 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅𝑧𝐴))
1413pm4.71rd 562 . . . 4 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅 ↔ (𝑧𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑅)))
156, 7elimasn 6110 . . . . 5 (𝑦 ∈ (𝑅 “ {𝑧}) ↔ ⟨𝑧, 𝑦⟩ ∈ 𝑅)
1615anbi2i 623 . . . 4 ((𝑧𝐴𝑦 ∈ (𝑅 “ {𝑧})) ↔ (𝑧𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑅))
1714, 16bitr4di 289 . . 3 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅 ↔ (𝑧𝐴𝑦 ∈ (𝑅 “ {𝑧}))))
18 sneq 4641 . . . . 5 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1918imaeq2d 6080 . . . 4 (𝑥 = 𝑧 → (𝑅 “ {𝑥}) = (𝑅 “ {𝑧}))
2019opeliunxp2 5852 . . 3 (⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})) ↔ (𝑧𝐴𝑦 ∈ (𝑅 “ {𝑧})))
2117, 20bitr4di 289 . 2 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥}))))
221, 5, 21eqrelrdv 5805 1 (ran 𝑅𝐴𝑅 = 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  wss 3963  {csn 4631  cop 4637   ciun 4996   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-iun 4998  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  gsummpt2co  33034
  Copyright terms: Public domain W3C validator