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Theorem bnj564 31121
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj564.17 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
Assertion
Ref Expression
bnj564 (𝜏 → dom 𝑓 = 𝑚)

Proof of Theorem bnj564
StepHypRef Expression
1 bnj564.17 . . 3 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
21simp1bi 1140 . 2 (𝜏𝑓 Fn 𝑚)
3 fndm 6151 . 2 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
42, 3syl 17 1 (𝜏 → dom 𝑓 = 𝑚)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072   = wceq 1632  dom cdm 5266   Fn wfn 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-an 385  df-3an 1074  df-fn 6052
This theorem is referenced by:  bnj570  31282  bnj916  31310
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