Theorem feq23 3623

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 3583	. . 3 |- (A = C -> (F Fn A <-> F Fn C))
2		sseq2 2083	. . 3 |- (B = D -> (ran F (_ B <-> ran F (_ D))
3	1, 2	bi2anan9 632	. 2 |- ((A = C /\ B = D) -> ((F Fn A /\ ran F (_ B) <-> (F Fn C /\ ran F (_ D)))
4		df-f 3194	. 2 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
5		df-f 3194	. 2 |- (F:C-->D <-> (F Fn C /\ ran F (_ D))
6	3, 4, 5	3bitr4g 555	1 |- ((A = C /\ B = D) -> (F:A-->B <-> F:C-->D))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   (_ wss 2047  ran crn 3171   Fn wfn 3177  -->wf 3178

This theorem is referenced by:  metcnp 7887  metcn 7889  cncfmet 7905  elghom 10384  mapdiscn 10511

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

Copyright terms: Public domain