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Theorem dfcnv2 29797
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 5707 . 2 Rel 𝑅
2 relxp 5322 . . . 4 Rel ({𝑥} × (𝑅 “ {𝑥}))
32rgenw 3108 . . 3 𝑥𝐴 Rel ({𝑥} × (𝑅 “ {𝑥}))
4 reliun 5435 . . 3 (Rel 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})) ↔ ∀𝑥𝐴 Rel ({𝑥} × (𝑅 “ {𝑥})))
53, 4mpbir 222 . 2 Rel 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥}))
6 vex 3390 . . . . . . . . 9 𝑧 ∈ V
7 vex 3390 . . . . . . . . 9 𝑦 ∈ V
86, 7opeldm 5523 . . . . . . . 8 (⟨𝑧, 𝑦⟩ ∈ 𝑅𝑧 ∈ dom 𝑅)
9 df-rn 5316 . . . . . . . 8 ran 𝑅 = dom 𝑅
108, 9syl6eleqr 2892 . . . . . . 7 (⟨𝑧, 𝑦⟩ ∈ 𝑅𝑧 ∈ ran 𝑅)
11 ssel2 3787 . . . . . . 7 ((ran 𝑅𝐴𝑧 ∈ ran 𝑅) → 𝑧𝐴)
1210, 11sylan2 582 . . . . . 6 ((ran 𝑅𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑅) → 𝑧𝐴)
1312ex 399 . . . . 5 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅𝑧𝐴))
1413pm4.71rd 554 . . . 4 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅 ↔ (𝑧𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑅)))
156, 7elimasn 5694 . . . . 5 (𝑦 ∈ (𝑅 “ {𝑧}) ↔ ⟨𝑧, 𝑦⟩ ∈ 𝑅)
1615anbi2i 611 . . . 4 ((𝑧𝐴𝑦 ∈ (𝑅 “ {𝑧})) ↔ (𝑧𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑅))
1714, 16syl6bbr 280 . . 3 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅 ↔ (𝑧𝐴𝑦 ∈ (𝑅 “ {𝑧}))))
18 sneq 4374 . . . . 5 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1918imaeq2d 5670 . . . 4 (𝑥 = 𝑧 → (𝑅 “ {𝑥}) = (𝑅 “ {𝑧}))
2019opeliunxp2 5456 . . 3 (⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})) ↔ (𝑧𝐴𝑦 ∈ (𝑅 “ {𝑧})))
2117, 20syl6bbr 280 . 2 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥}))))
221, 5, 21eqrelrdv 5412 1 (ran 𝑅𝐴𝑅 = 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2155  wral 3092  wss 3763  {csn 4364  cop 4370   ciun 4705   × cxp 5303  ccnv 5304  dom cdm 5305  ran crn 5306  cima 5308  Rel wrel 5310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pr 5090
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-iun 4707  df-br 4838  df-opab 4900  df-xp 5311  df-rel 5312  df-cnv 5313  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318
This theorem is referenced by:  gsummpt2co  30099
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