Theorem funss 5274

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 4747	. . 3 |- ( A C_ B -> ( Rel B -> Rel A ) )
2		coss1 4818	. . . . 5 |- ( A C_ B -> ( A o. `' A ) C_ ( B o. `' A ) )
3		cnvss 4836	. . . . . 6 |- ( A C_ B -> `' A C_ `' B )
4		coss2 4819	. . . . . 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 3190	. . . 4 |- ( A C_ B -> ( A o. `' A ) C_ ( B o. `' B ) )
7		sstr2 3187	. . . 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 5257	. 2 |- ( Fun B <-> ( Rel B /\ ( B o. `' B ) C_ _I ) )
11		df-fun 5257	. 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 3154   _I cid 4320   `'ccnv 4659   o. ccom 4664   Rel wrel 4665   Fun wfun 5249

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3160  df-ss 3167  df-br 4031  df-opab 4092  df-rel 4667  df-cnv 4668  df-co 4669  df-fun 5257

This theorem is referenced by:  funeq  5275  funopab4  5292  funres  5296  fun0  5313  funcnvcnv  5314  funin  5326  funres11  5327  foimacnv  5519  tfrlemibfn  6383  tfr1onlembfn  6399  tfrcllembfn  6412  strslssd  12668  strle1g  12727