Theorem relbrcnvg 6096

Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5856 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
relbrcnvg	⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Proof of Theorem relbrcnvg

Step	Hyp	Ref	Expression
1		relcnv 6095	. . . 4 ⊢ Rel ◡𝑅
2	1	brrelex12i 5704	. . 3 ⊢ (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	a1i 11	. 2 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
4		brrelex12 5701	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
5	4	ancomd 465	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	5	ex 416	. 2 ⊢ (Rel 𝑅 → (𝐵𝑅𝐴 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
7		brcnvg 5853	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))
8	7	a1i 11	. 2 ⊢ (Rel 𝑅 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)))
9	3, 6, 8	pm5.21ndd 381	1 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2144  Vcvv 3456   class class class wbr 5102  ^◡ccnv 5648  Rel wrel 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657

This theorem is referenced by:  eliniseg2  6097  relbrcnv  6098  isinv  17795  releleccnv  38764  relcnveq2  38833  elrelscnveq2  39133  eqvrelsym  39193  brco2f1o  44613  brco3f1o  44614  ntrclsnvobr  44633  neicvgel1  44700