Theorem releccnveq 38242

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 2723	. 2 ⊢ ([𝐴]◡𝑅 = [𝐵]◡𝑆 ↔ ∀𝑥(𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆))
2		releleccnv 38241	. . . 4 ⊢ (Rel 𝑅 → (𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥𝑅𝐴))
3		releleccnv 38241	. . . 4 ⊢ (Rel 𝑆 → (𝑥 ∈ [𝐵]◡𝑆 ↔ 𝑥𝑆𝐵))
4	2, 3	bi2bian9 640	. . 3 ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → ((𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆) ↔ (𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵)))
5	4	albidv 1920	. 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 1538   = wceq 1540   ∈ wcel 2109   class class class wbr 5109  ^◡ccnv 5639  Rel wrel 5645  [cec 8671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-xp 5646  df-rel 5647  df-cnv 5648  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-ec 8675

This theorem is referenced by:  extssr  38495