Theorem feq23 3629

Description: Equality theorem for functions. (Contributed by FL, 14-Jul-2007.)

Assertion

Ref	Expression
feq23	|- ((A = C /\ B = D) -> (F:A-->B <-> F:C-->D))

Proof of Theorem feq23

Step	Hyp	Ref	Expression
1		fneq2 3589	. . 3 |- (A = C -> (F Fn A <-> F Fn C))
2		sseq2 2086	. . 3 |- (B = D -> (ran F (_ B <-> ran F (_ D))
3	1, 2	bi2anan9 634	. 2 |- ((A = C /\ B = D) -> ((F Fn A /\ ran F (_ B) <-> (F Fn C /\ ran F (_ D)))
4		df-f 3200	. 2 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
5		df-f 3200	. 2 |- (F:C-->D <-> (F Fn C /\ ran F (_ D))
6	3, 4, 5	3bitr4g 557	1 |- ((A = C /\ B = D) -> (F:A-->B <-> F:C-->D))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   (_ wss 2050  ran crn 3177   Fn wfn 3183  -->wf 3184

This theorem is referenced by:  metcnp 7884  metcn 7886  cncfmet 7902  elghom 10379  mapdiscn 10497

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056  df-fn 3199  df-f 3200

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