Theorem funss 5137

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 4621	. . 3 |- ( A C_ B -> ( Rel B -> Rel A ) )
2		coss1 4689	. . . . 5 |- ( A C_ B -> ( A o. `' A ) C_ ( B o. `' A ) )
3		cnvss 4707	. . . . . 6 |- ( A C_ B -> `' A C_ `' B )
4		coss2 4690	. . . . . 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 3102	. . . 4 |- ( A C_ B -> ( A o. `' A ) C_ ( B o. `' B ) )
7		sstr2 3099	. . . 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 333	. 2 |- ( A C_ B -> ( ( Rel B /\ ( B o. `' B ) C_ _I ) -> ( Rel A /\ ( A o. `' A ) C_ _I ) ) )
10		df-fun 5120	. 2 |- ( Fun B <-> ( Rel B /\ ( B o. `' B ) C_ _I ) )
11		df-fun 5120	. 2 |- ( Fun A <-> ( Rel A /\ ( A o. `' A ) C_ _I ) )
12	9, 10, 11	3imtr4g 204	1 |- ( A C_ B -> ( Fun B -> Fun A ) )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3066   _I cid 4205   `'ccnv 4533   o. ccom 4538   Rel wrel 4539   Fun wfun 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079  df-br 3925  df-opab 3985  df-rel 4541  df-cnv 4542  df-co 4543  df-fun 5120

This theorem is referenced by:  funeq  5138  funopab4  5155  funres  5159  fun0  5176  funcnvcnv  5177  funin  5189  funres11  5190  foimacnv  5378  tfrlemibfn  6218  tfr1onlembfn  6234  tfrcllembfn  6247  strslssd  11994  strle1g  12038