Theorem funss 5290

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 4762	. . 3 |- ( A C_ B -> ( Rel B -> Rel A ) )
2		coss1 4833	. . . . 5 |- ( A C_ B -> ( A o. `' A ) C_ ( B o. `' A ) )
3		cnvss 4851	. . . . . 6 |- ( A C_ B -> `' A C_ `' B )
4		coss2 4834	. . . . . 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 3203	. . . 4 |- ( A C_ B -> ( A o. `' A ) C_ ( B o. `' B ) )
7		sstr2 3200	. . . 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 335	. 2 |- ( A C_ B -> ( ( Rel B /\ ( B o. `' B ) C_ _I ) -> ( Rel A /\ ( A o. `' A ) C_ _I ) ) )
10		df-fun 5273	. 2 |- ( Fun B <-> ( Rel B /\ ( B o. `' B ) C_ _I ) )
11		df-fun 5273	. 2 |- ( Fun A <-> ( Rel A /\ ( A o. `' A ) C_ _I ) )
12	9, 10, 11	3imtr4g 205	1 |- ( A C_ B -> ( Fun B -> Fun A ) )

Syntax hints:   -> wi 4   /\ wa 104   C_ wss 3166   _I cid 4335   `'ccnv 4674   o. ccom 4679   Rel wrel 4680   Fun wfun 5265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179  df-br 4045  df-opab 4106  df-rel 4682  df-cnv 4683  df-co 4684  df-fun 5273

This theorem is referenced by:  funeq  5291  funopab4  5308  funres  5312  fun0  5332  funcnvcnv  5333  funin  5345  funres11  5346  foimacnv  5540  tfrlemibfn  6414  tfr1onlembfn  6430  tfrcllembfn  6443  strslssd  12879  strle1g  12938