Theorem relbrcnvg 6112

Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5866 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 6111	. . . 4 ⊢ Rel ◡𝑅
2	1	brrelex12i 5710	. . 3 ⊢ (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	a1i 11	. 2 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
4		brrelex12 5707	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
5	4	ancomd 467	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	5	ex 418	. 2 ⊢ (Rel 𝑅 → (𝐵𝑅𝐴 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
7		brcnvg 5863	. . 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 382	1 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106  ^◡ccnv 5654  Rel wrel 5660

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663

This theorem is used by:  eliniseg2  6113  relbrcnv  6114  isinv  17961  releleccnv  39383  relcnveq2  39452  elrelscnveq2  39752  eqvrelsym  39812  brco2f1o  45222  brco3f1o  45223  ntrclsnvobr  45242  neicvgel1  45309