Theorem funss 4940

Description: Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)

Assertion

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

Proof of Theorem funss

Step	Hyp	Ref	Expression
1		relss 4445	. . 3 |- ( A C_ B -> ( Rel B -> Rel A ) )
2		coss1 4509	. . . . 5 |- ( A C_ B -> ( A o. `' A ) C_ ( B o. `' A ) )
3		cnvss 4526	. . . . . 6 |- ( A C_ B -> `' A C_ `' B )
4		coss2 4510	. . . . . 6 |- ( `' A C_ `' B -> ( B o. `' A ) C_ ( B o. `' B ) )
5	3, 4	syl 14	. . . . 5 |- ( A C_ B -> ( B o. `' A ) C_ ( B o. `' B ) )
6	2, 5	sstrd 3009	. . . 4 |- ( A C_ B -> ( A o. `' A ) C_ ( B o. `' B ) )
7		sstr2 3006	. . . 4 |- ( ( A o. `' A ) C_ ( B o. `' B ) -> ( ( B o. `' B ) C_ _I -> ( A o. `' A ) C_ _I ) )
8	6, 7	syl 14	. . 3 |- ( A C_ B -> ( ( B o. `' B ) C_ _I -> ( A o. `' A ) C_ _I ) )
9	1, 8	anim12d 328	. 2 |- ( A C_ B -> ( ( Rel B /\ ( B o. `' B ) C_ _I ) -> ( Rel A /\ ( A o. `' A ) C_ _I ) ) )
10		df-fun 4924	. 2 |- ( Fun B <-> ( Rel B /\ ( B o. `' B ) C_ _I ) )
11		df-fun 4924	. 2 |- ( Fun A <-> ( Rel A /\ ( A o. `' A ) C_ _I ) )
12	9, 10, 11	3imtr4g 203	1 |- ( A C_ B -> ( Fun B -> Fun A ) )

Syntax hints:   -> wi 4   /\ wa 102   C_ wss 2973   _I cid 4043   `'ccnv 4362   o. ccom 4367   Rel wrel 4368   Fun wfun 4916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-in 2979  df-ss 2986  df-br 3786  df-opab 3840  df-rel 4370  df-cnv 4371  df-co 4372  df-fun 4924

This theorem is referenced by:  funeq  4941  funopab4  4957  funres  4961  fun0  4977  funcnvcnv  4978  funin  4990  funres11  4991  foimacnv  5164  tfrlemibfn  5965