Theorem brfvid 44117

Description: If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020.)

Hypothesis

Ref	Expression
brfvid.r	⊢ (𝜑 → 𝑅 ∈ V)

Assertion

Ref	Expression
brfvid	⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))

Proof of Theorem brfvid

Step	Hyp	Ref	Expression
1		brfvid.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
2		fvi 6908	. . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅)
4	3	breqd 5097	1 ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3430   class class class wbr 5086   I cid 5516  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498

This theorem is referenced by: (None)