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Theorem funss 5399

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 4864	. . 3
2		coss1 4937	. . . . 5
3		cnvss 4955	. . . . . 6
4		coss2 4938	. . . . . 6
5	3, 4	syl 14	. . . . 5
6	2, 5	sstrd 3258	. . . 4
7		sstr2 3255	. . . 4
8	6, 7	syl 14	. . 3
9	1, 8	anim12d 335	. 2 $/\$ $/\$
10		df-fun 5382	. 2 $/\$
11		df-fun 5382	. 2 $/\$
12	9, 10, 11	3imtr4g 205	1

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wss 3220

cid 4435

ccnv 4775

ccom 4780

wrel 4781

wfun 5374

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-ext 2220

This proof depends on definitions: df-bi 117 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-in 3226 df-ss 3233 df-br 4132 df-opab 4194 df-rel 4783 df-cnv 4784 df-co 4785 df-fun 5382

This theorem is used by: funeq 5400 funopab4 5417 funres 5421 fun0 5442 funcnvcnv 5443 funin 5455 funres11 5456 foimacnv 5660 funsssuppss 6502 tfrlemibfn 6603 tfr1onlembfn 6619 tfrcllembfn 6632 strslssd 13457 strle1g 13519 subgrfun 16713