Theorem fss 5231

Description: Expanding the codomain of a mapping. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 10-May-1998.) (Revised by set.mm contributors, 18-Sep-2011.)

Assertion

Ref	Expression
fss	|- ((F:A-->B /\ B C_ C) -> F:A-->C)

Proof of Theorem fss

Step	Hyp	Ref	Expression
1		sstr2 3280	. . . . 5 |- (ran F C_ B -> (B C_ C -> ran F C_ C))
2	1	com12 27	. . . 4 |- (B C_ C -> (ran F C_ B -> ran F C_ C))
3	2	anim2d 548	. . 3 |- (B C_ C -> ((F Fn A /\ ran F C_ B) -> (F Fn A /\ ran F C_ C)))
4		df-f 4792	. . 3 |- (F:A-->B <-> (F Fn A /\ ran F C_ B))
5		df-f 4792	. . 3 |- (F:A-->C <-> (F Fn A /\ ran F C_ C))
6	3, 4, 5	3imtr4g 261	. 2 |- (B C_ C -> (F:A-->B -> F:A-->C))
7	6	impcom 419	1 |- ((F:A-->B /\ B C_ C) -> F:A-->C)

Syntax hints:   -> wi 4   /\ wa 358   C_ wss 3258  ran crn 4774   Fn wfn 4777  -->wf 4778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-f 4792

This theorem is referenced by:  f1ss  5263  ffoss  5315  fsn2  5435  map0  6026  mapsn  6027  mapss  6028