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Theorem releccnveq 34365
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 2765 . 2 ([𝐴]𝑅 = [𝐵]𝑆 ↔ ∀𝑥(𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑆))
2 releleccnv 34364 . . . 4 (Rel 𝑅 → (𝑥 ∈ [𝐴]𝑅𝑥𝑅𝐴))
3 releleccnv 34364 . . . 4 (Rel 𝑆 → (𝑥 ∈ [𝐵]𝑆𝑥𝑆𝐵))
42, 3bi2bian9 622 . . 3 ((Rel 𝑅 ∧ Rel 𝑆) → ((𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑆) ↔ (𝑥𝑅𝐴𝑥𝑆𝐵)))
54albidv 2001 . 2 ((Rel 𝑅 ∧ Rel 𝑆) → (∀𝑥(𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑆) ↔ ∀𝑥(𝑥𝑅𝐴𝑥𝑆𝐵)))
61, 5syl5bb 272 1 ((Rel 𝑅 ∧ Rel 𝑆) → ([𝐴]𝑅 = [𝐵]𝑆 ↔ ∀𝑥(𝑥𝑅𝐴𝑥𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wcel 2145   class class class wbr 4786  ccnv 5248  Rel wrel 5254  [cec 7894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ec 7898
This theorem is referenced by:  extssr  34601
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