Theorem bnj1422 30908

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

Hypotheses

Ref	Expression
bnj1422.1	|- ( ph -> Fun A )
bnj1422.2	|- ( ph -> dom A = B )

Assertion

Ref	Expression
bnj1422	|- ( ph -> A Fn B )

Proof of Theorem bnj1422

Step	Hyp	Ref	Expression
1		bnj1422.1	. 2 |- ( ph -> Fun A )
2		bnj1422.2	. 2 |- ( ph -> dom A = B )
3		df-fn 5891	. 2 |- ( A Fn B <-> ( Fun A /\ dom A = B ) )
4	1, 2, 3	sylanbrc 698	1 |- ( ph -> A Fn B )

Syntax hints:   -> wi 4   = wceq 1483   dom cdm 5114   Fun wfun 5882   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-fn 5891

This theorem is referenced by:  bnj150  30946  bnj535  30960  bnj1312  31126  bnj60  31130