Theorem relbrcnvg 6135

Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5907 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 6134	. . . 4 ⊢ Rel ◡𝑅
2	1	brrelex12i 5755	. . 3 ⊢ (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	a1i 11	. 2 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
4		brrelex12 5752	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
5	4	ancomd 461	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	5	ex 412	. 2 ⊢ (Rel 𝑅 → (𝐵𝑅𝐴 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
7		brcnvg 5904	. . 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 379	1 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  ^◡ccnv 5699  Rel wrel 5705

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-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-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-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

This theorem is referenced by:  eliniseg2  6136  relbrcnv  6137  isinv  17821  releleccnv  38213  relcnveq2  38279  elrelscnveq2  38449  eqvrelsym  38561  brco2f1o  43994  brco3f1o  43995  ntrclsnvobr  44014  neicvgel1  44081