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Theorem bnj1422 31215
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1422.1 (𝜑 → Fun 𝐴)
bnj1422.2 (𝜑 → dom 𝐴 = 𝐵)
Assertion
Ref Expression
bnj1422 (𝜑𝐴 Fn 𝐵)

Proof of Theorem bnj1422
StepHypRef Expression
1 bnj1422.1 . 2 (𝜑 → Fun 𝐴)
2 bnj1422.2 . 2 (𝜑 → dom 𝐴 = 𝐵)
3 df-fn 6052 . 2 (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
41, 2, 3sylanbrc 701 1 (𝜑𝐴 Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  dom cdm 5266  Fun wfun 6043   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-fn 6052
This theorem is referenced by:  bnj150  31253  bnj535  31267  bnj1312  31433  bnj60  31437
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