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Theorem releccnveq 35672
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 2795 . 2 ([𝐴]𝑅 = [𝐵]𝑆 ↔ ∀𝑥(𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑆))
2 releleccnv 35671 . . . 4 (Rel 𝑅 → (𝑥 ∈ [𝐴]𝑅𝑥𝑅𝐴))
3 releleccnv 35671 . . . 4 (Rel 𝑆 → (𝑥 ∈ [𝐵]𝑆𝑥𝑆𝐵))
42, 3bi2bian9 640 . . 3 ((Rel 𝑅 ∧ Rel 𝑆) → ((𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑆) ↔ (𝑥𝑅𝐴𝑥𝑆𝐵)))
54albidv 1921 . 2 ((Rel 𝑅 ∧ Rel 𝑆) → (∀𝑥(𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑆) ↔ ∀𝑥(𝑥𝑅𝐴𝑥𝑆𝐵)))
61, 5syl5bb 286 1 ((Rel 𝑅 ∧ Rel 𝑆) → ([𝐴]𝑅 = [𝐵]𝑆 ↔ ∀𝑥(𝑥𝑅𝐴𝑥𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2112   class class class wbr 5033  ccnv 5522  Rel wrel 5528  [cec 8274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ec 8278
This theorem is referenced by:  extssr  35902
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