Theorem bnj564 32019

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 1141	. 2 ⊢ (𝜏 → 𝑓 Fn 𝑚)
3		fndm 6458	. 2 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
4	2, 3	syl 17	1 ⊢ (𝜏 → dom 𝑓 = 𝑚)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1536  dom cdm 5558   Fn wfn 6353

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 209  df-an 399  df-3an 1085  df-fn 6361

This theorem is referenced by:  bnj570  32181  bnj916  32209