Theorem bnj1422 30162

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	⊢ (𝜑 → 𝐴 Fn 𝐵)

Proof of Theorem bnj1422

Step	Hyp	Ref	Expression
1		bnj1422.1	. 2 ⊢ (𝜑 → Fun 𝐴)
2		bnj1422.2	. 2 ⊢ (𝜑 → dom 𝐴 = 𝐵)
3		df-fn 5807	. 2 ⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
4	1, 2, 3	sylanbrc 695	1 ⊢ (𝜑 → 𝐴 Fn 𝐵)

Syntax hints:   → wi 4   = wceq 1475  dom cdm 5038  Fun wfun 5798   Fn wfn 5799

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 196  df-an 385  df-fn 5807

This theorem is referenced by:  bnj150  30200  bnj535  30214  bnj1312  30380  bnj60  30384