Theorem releccnveq 38970

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 2758	. 2 ⊢ ([𝐴]◡𝑅 = [𝐵]◡𝑆 ↔ ∀𝑥(𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆))
2		releleccnv 38969	. . . 4 ⊢ (Rel 𝑅 → (𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥𝑅𝐴))
3		releleccnv 38969	. . . 4 ⊢ (Rel 𝑆 → (𝑥 ∈ [𝐵]◡𝑆 ↔ 𝑥𝑆𝐵))
4	2, 3	bi2bian9 652	. . 3 ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → ((𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆) ↔ (𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵)))
5	4	albidv 1953	. 2 ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → (∀𝑥(𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆) ↔ ∀𝑥(𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵)))
6	1, 5	bitrid 286	1 ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → ([𝐴]◡𝑅 = [𝐵]◡𝑆 ↔ ∀𝑥(𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146   class class class wbr 5111  ^◡ccnv 5662  Rel wrel 5668  [cec 8698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8702

This theorem is used by:  extssr  39298