Theorem bnj564 30814

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj564.17	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )

Assertion

Ref	Expression
bnj564	|- ( ta -> dom f = m )

Proof of Theorem bnj564

Step	Hyp	Ref	Expression
1		bnj564.17	. . 3 |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
2	1	simp1bi 1076	. 2 |- ( ta -> f Fn m )
3		fndm 5990	. 2 |- ( f Fn m -> dom f = m )
4	2, 3	syl 17	1 |- ( ta -> dom f = m )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   dom cdm 5114   Fn wfn 5883

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 386  df-3an 1039  df-fn 5891

This theorem is referenced by:  bnj570  30975  bnj916  31003