Theorem releccnveq 38599

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(𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆

Proof of Theorem releccnveq

Step	Hyp	Ref	Expression
1		dfcleq 2730	. 2 ⊢ ([𝐴]◡𝑅 = [𝐵]◡𝑆 ↔ ∀𝑥(𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆))
2		releleccnv 38598	. . . 4 ⊢ (Rel 𝑅 → (𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥𝑅𝐴))
3		releleccnv 38598	. . . 4 ⊢ (Rel 𝑆 → (𝑥 ∈ [𝐵]◡𝑆 ↔ 𝑥𝑆𝐵))
4	2, 3	bi2bian9 641	. . 3 ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → ((𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆) ↔ (𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵)))
5	4	albidv 1922	. 2 ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → (∀𝑥(𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆) ↔ ∀𝑥(𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵)))
6	1, 5	bitrid 283	1 ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → ([𝐴]◡𝑅 = [𝐵]◡𝑆 ↔ ∀𝑥(𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ^◡ccnv 5624  Rel wrel 5630  [cec 8635

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639

This theorem is referenced by:  extssr  38927