Theorem funss 4439

Description: Subclass theorem for function predicate. (The proof was shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
funss	|- (A C_ B -> (Fun B -> Fun A))

Proof of Theorem funss

Step	Hyp	Ref	Expression
1		relss 4074	. . 3 |- (A C_ B -> (Rel B -> Rel A))
2		id 73	. . . . . . 7 |- (A C_ B -> A C_ B)
3	2	ssbrd 3378	. . . . . 6 |- (A C_ B -> (xAy -> xBy))
4	3	19.21aiv 1664	. . . . 5 |- (A C_ B -> A.y(xAy -> xBy))
5		immo 1813	. . . . 5 |- (A.y(xAy -> xBy) -> (E*y xBy -> E*y xAy))
6	4, 5	syl 12	. . . 4 |- (A C_ B -> (E*y xBy -> E*y xAy))
7	6	alimdv 1668	. . 3 |- (A C_ B -> (A.xE*y xBy -> A.xE*y xAy))
8	1, 7	anim12d 617	. 2 |- (A C_ B -> ((Rel B /\ A.xE*y xBy) -> (Rel A /\ A.xE*y xAy)))
9		dffun6 4436	. 2 |- (Fun B <-> (Rel B /\ A.xE*y xBy))
10		dffun6 4436	. 2 |- (Fun A <-> (Rel A /\ A.xE*y xAy))
11	8, 9, 10	3imtr4g 612	1 |- (A C_ B -> (Fun B -> Fun A))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296  E*wmo 1772   C_ wss 2593   class class class wbr 3338  Rel wrel 3991  Fun wfun 3992

This theorem is referenced by:  funeq 4441  funres 4459  fun0 4472  funcnvcnv 4473  funin 4484  funres11 4486  foimacnv 4657  fodom 5960  subtopmetlem 10255  limfil 10297  isfilmap 10308  sflimf 10318  cmpfun 14480  dualalg 15131  sfcls 15604  filclus 15605  sfclusf 15624  tailf 15633  istail 15634

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-rel 4001  df-cnv 4002  df-co 4003  df-fun 4008

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