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(𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵)))

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

Proof of Theorem releccnveq

Step	Hyp	Ref	Expression
1		dfcleq 2733	. 2 ⊢ ([𝐴]◡𝑅 = [𝐵]◡𝑆 ↔ ∀𝑥(𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆))
2		releleccnv 38215	. . . 4 ⊢ (Rel 𝑅 → (𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥𝑅𝐴))
3		releleccnv 38215	. . . 4 ⊢ (Rel 𝑆 → (𝑥 ∈ [𝐵]◡𝑆 ↔ 𝑥𝑆𝐵))
4	2, 3	bi2bian9 639	. . 3 ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → ((𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆) ↔ (𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵)))
5	4	albidv 1919	. 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 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