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Theorem releccnveq 38216
Description: Equality of converse 𝑅-coset and converse 𝑆-coset when 𝑅 and 𝑆 are relations. (Contributed by Peter Mazsa, 27-Jul-2019.)
Assertion
Ref Expression
releccnveq ((Rel 𝑅 ∧ Rel 𝑆) → ([𝐴]𝑅 = [𝐵]𝑆 ↔ ∀𝑥(𝑥𝑅𝐴𝑥𝑆𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆

Proof of Theorem releccnveq
StepHypRef Expression
1 dfcleq 2733 . 2 ([𝐴]𝑅 = [𝐵]𝑆 ↔ ∀𝑥(𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑆))
2 releleccnv 38215 . . . 4 (Rel 𝑅 → (𝑥 ∈ [𝐴]𝑅𝑥𝑅𝐴))
3 releleccnv 38215 . . . 4 (Rel 𝑆 → (𝑥 ∈ [𝐵]𝑆𝑥𝑆𝐵))
42, 3bi2bian9 639 . . 3 ((Rel 𝑅 ∧ Rel 𝑆) → ((𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑆) ↔ (𝑥𝑅𝐴𝑥𝑆𝐵)))
54albidv 1919 . 2 ((Rel 𝑅 ∧ Rel 𝑆) → (∀𝑥(𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑆) ↔ ∀𝑥(𝑥𝑅𝐴𝑥𝑆𝐵)))
61, 5bitrid 283 1 ((Rel 𝑅 ∧ Rel 𝑆) → ([𝐴]𝑅 = [𝐵]𝑆 ↔ ∀𝑥(𝑥𝑅𝐴𝑥𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2108   class class class wbr 5166  ccnv 5699  Rel wrel 5705  [cec 8763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8767
This theorem is referenced by:  extssr  38467
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