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Theorem dfcnv2 29604
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 5538 . 2 Rel 𝑅
2 relxp 5160 . . . 4 Rel ({𝑥} × (𝑅 “ {𝑥}))
32rgenw 2953 . . 3 𝑥𝐴 Rel ({𝑥} × (𝑅 “ {𝑥}))
4 reliun 5272 . . 3 (Rel 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})) ↔ ∀𝑥𝐴 Rel ({𝑥} × (𝑅 “ {𝑥})))
53, 4mpbir 221 . 2 Rel 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥}))
6 vex 3234 . . . . . . . . 9 𝑧 ∈ V
7 vex 3234 . . . . . . . . 9 𝑦 ∈ V
86, 7opeldm 5360 . . . . . . . 8 (⟨𝑧, 𝑦⟩ ∈ 𝑅𝑧 ∈ dom 𝑅)
9 df-rn 5154 . . . . . . . 8 ran 𝑅 = dom 𝑅
108, 9syl6eleqr 2741 . . . . . . 7 (⟨𝑧, 𝑦⟩ ∈ 𝑅𝑧 ∈ ran 𝑅)
11 ssel2 3631 . . . . . . 7 ((ran 𝑅𝐴𝑧 ∈ ran 𝑅) → 𝑧𝐴)
1210, 11sylan2 490 . . . . . 6 ((ran 𝑅𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑅) → 𝑧𝐴)
1312ex 449 . . . . 5 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅𝑧𝐴))
1413pm4.71rd 668 . . . 4 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅 ↔ (𝑧𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑅)))
156, 7elimasn 5525 . . . . 5 (𝑦 ∈ (𝑅 “ {𝑧}) ↔ ⟨𝑧, 𝑦⟩ ∈ 𝑅)
1615anbi2i 730 . . . 4 ((𝑧𝐴𝑦 ∈ (𝑅 “ {𝑧})) ↔ (𝑧𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑅))
1714, 16syl6bbr 278 . . 3 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅 ↔ (𝑧𝐴𝑦 ∈ (𝑅 “ {𝑧}))))
18 sneq 4220 . . . . 5 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1918imaeq2d 5501 . . . 4 (𝑥 = 𝑧 → (𝑅 “ {𝑥}) = (𝑅 “ {𝑧}))
2019opeliunxp2 5293 . . 3 (⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})) ↔ (𝑧𝐴𝑦 ∈ (𝑅 “ {𝑧})))
2117, 20syl6bbr 278 . 2 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥}))))
221, 5, 21eqrelrdv 5250 1 (ran 𝑅𝐴𝑅 = 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wss 3607  {csn 4210  cop 4216   ciun 4552   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-iun 4554  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  gsummpt2co  29908
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