Theorem brfvid 41184

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 6826	. . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅)
4	3	breqd 5081	1 ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070   I cid 5479  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426

This theorem is referenced by: (None)