Theorem funss 5371

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 4837	. . 3 |- ( A C_ B -> ( Rel B -> Rel A ) )
2		coss1 4910	. . . . 5 |- ( A C_ B -> ( A o. `' A ) C_ ( B o. `' A ) )
3		cnvss 4928	. . . . . 6 |- ( A C_ B -> `' A C_ `' B )
4		coss2 4911	. . . . . 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 3248	. . . 4 |- ( A C_ B -> ( A o. `' A ) C_ ( B o. `' B ) )
7		sstr2 3245	. . . 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 5354	. 2 |- ( Fun B <-> ( Rel B /\ ( B o. `' B ) C_ _I ) )
11		df-fun 5354	. 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 3211   _I cid 4409   `'ccnv 4748   o. ccom 4753   Rel wrel 4754   Fun wfun 5346

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-in 3217  df-ss 3224  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-fun 5354

This theorem is referenced by:  funeq  5372  funopab4  5389  funres  5393  fun0  5414  funcnvcnv  5415  funin  5427  funres11  5428  foimacnv  5632  funsssuppss  6458  tfrlemibfn  6559  tfr1onlembfn  6575  tfrcllembfn  6588  strslssd  13259  strle1g  13319  subgrfun  16262