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Theorem brfvid 38757
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 (𝜑𝑅 ∈ V)
Assertion
Ref Expression
brfvid (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))

Proof of Theorem brfvid
StepHypRef Expression
1 brfvid.r . . 3 (𝜑𝑅 ∈ V)
2 fvi 6481 . . 3 (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
31, 2syl 17 . 2 (𝜑 → ( I ‘𝑅) = 𝑅)
43breqd 4855 1 (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  Vcvv 3386   class class class wbr 4844   I cid 5220  cfv 6102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110
This theorem is referenced by: (None)
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