Theorem ffdm 3639

Description: A mapping is a partial function.

Assertion

Ref	Expression
ffdm	|- (F:A-->B -> (F:dom F-->B /\ dom F (_ A))

Proof of Theorem ffdm

Step	Hyp	Ref	Expression
1		fdm 3631	. . . 4 |- (F:A-->B -> dom F = A)
2		feq2 3621	. . . 4 |- (dom F = A -> (F:dom F-->B <-> F:A-->B))
3	1, 2	syl 10	. . 3 |- (F:A-->B -> (F:dom F-->B <-> F:A-->B))
4	3	ibir 593	. 2 |- (F:A-->B -> F:dom F-->B)
5		eqimss 2109	. . 3 |- (dom F = A -> dom F (_ A)
6	1, 5	syl 10	. 2 |- (F:A-->B -> dom F (_ A)
7	4, 6	jca 288	1 |- (F:A-->B -> (F:dom F-->B /\ dom F (_ A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   (_ wss 2047  dom cdm 3170  -->wf 3178

This theorem is referenced by:  fpm 4338

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053  df-fn 3193  df-f 3194

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