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Theorem bnj1422 32234
 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 6328 . 2 (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
41, 2, 3sylanbrc 586 1 (𝜑𝐴 Fn 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  dom cdm 5520  Fun wfun 6319   Fn wfn 6320 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 210  df-an 400  df-fn 6328 This theorem is referenced by:  bnj150  32273  bnj535  32287  bnj1312  32455  bnj60  32459
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