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Theorem funss 5150

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

**Assertion**
Ref	Expression
funss

**Proof of Theorem funss**
Step	Hyp	Ref	Expression
1		relss 4634	. . 3
2		coss1 4702	. . . . 5
3		cnvss 4720	. . . . . 6
4		coss2 4703	. . . . . 6
5	3, 4	syl 14	. . . . 5
6	2, 5	sstrd 3112	. . . 4
7		sstr2 3109	. . . 4
8	6, 7	syl 14	. . 3
9	1, 8	anim12d 333	. 2 $/\$ $/\$
10		df-fun 5133	. 2 $/\$
11		df-fun 5133	. 2 $/\$
12	9, 10, 11	3imtr4g 204	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wss 3076

cid 4218

ccnv 4546

ccom 4551

wrel 4552

wfun 5125

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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122

This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-in 3082 df-ss 3089 df-br 3938 df-opab 3998 df-rel 4554 df-cnv 4555 df-co 4556 df-fun 5133

This theorem is referenced by: funeq 5151 funopab4 5168 funres 5172 fun0 5189 funcnvcnv 5190 funin 5202 funres11 5203 foimacnv 5393 tfrlemibfn 6233 tfr1onlembfn 6249 tfrcllembfn 6262 strslssd 12044 strle1g 12088