Theorem bnj564 35375

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 𝑚 ∧ 𝜑′ ∧ 𝜓′))

Assertion

Ref	Expression
bnj564	⊢ (𝜏 → dom 𝑓 = 𝑚)

Proof of Theorem bnj564

Step	Hyp	Ref	Expression
1		bnj564.17	. . 3 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
2	1	simp1bi 1163	. 2 ⊢ (𝜏 → 𝑓 Fn 𝑚)
3	2	fndmd 6644	1 ⊢ (𝜏 → dom 𝑓 = 𝑚)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1569  dom cdm 5650   Fn wfn 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-fn 6541

This theorem is used by:  bnj570  35535  bnj916  35563