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Theorem dfcnv2 32406
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 6096 . 2 Rel 𝑅
2 relxp 5687 . . . 4 Rel ({𝑥} × (𝑅 “ {𝑥}))
32rgenw 3059 . . 3 𝑥𝐴 Rel ({𝑥} × (𝑅 “ {𝑥}))
4 reliun 5809 . . 3 (Rel 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})) ↔ ∀𝑥𝐴 Rel ({𝑥} × (𝑅 “ {𝑥})))
53, 4mpbir 230 . 2 Rel 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥}))
6 vex 3472 . . . . . . . . 9 𝑧 ∈ V
7 vex 3472 . . . . . . . . 9 𝑦 ∈ V
86, 7opeldm 5900 . . . . . . . 8 (⟨𝑧, 𝑦⟩ ∈ 𝑅𝑧 ∈ dom 𝑅)
9 df-rn 5680 . . . . . . . 8 ran 𝑅 = dom 𝑅
108, 9eleqtrrdi 2838 . . . . . . 7 (⟨𝑧, 𝑦⟩ ∈ 𝑅𝑧 ∈ ran 𝑅)
11 ssel2 3972 . . . . . . 7 ((ran 𝑅𝐴𝑧 ∈ ran 𝑅) → 𝑧𝐴)
1210, 11sylan2 592 . . . . . 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 6081 . . . . 5 (𝑦 ∈ (𝑅 “ {𝑧}) ↔ ⟨𝑧, 𝑦⟩ ∈ 𝑅)
1615anbi2i 622 . . . 4 ((𝑧𝐴𝑦 ∈ (𝑅 “ {𝑧})) ↔ (𝑧𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑅))
1714, 16bitr4di 289 . . 3 (ran 𝑅𝐴 → (⟨𝑧, 𝑦⟩ ∈ 𝑅 ↔ (𝑧𝐴𝑦 ∈ (𝑅 “ {𝑧}))))
18 sneq 4633 . . . . 5 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1918imaeq2d 6052 . . . 4 (𝑥 = 𝑧 → (𝑅 “ {𝑥}) = (𝑅 “ {𝑧}))
2019opeliunxp2 5831 . . 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 5785 1 (ran 𝑅𝐴𝑅 = 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  wss 3943  {csn 4623  cop 4629   ciun 4990   × cxp 5667  ccnv 5668  dom cdm 5669  ran crn 5670  cima 5672  Rel wrel 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-iun 4992  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682
This theorem is referenced by:  gsummpt2co  32704
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