funss - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > funss		Unicode version

Theorem funss 5396

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 4862	. . 3
2		coss1 4935	. . . . 5
3		cnvss 4953	. . . . . 6
4		coss2 4936	. . . . . 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 5379	. 2 $/\$
11		df-fun 5379	. 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 4433

ccnv 4773

ccom 4778

wrel 4779

wfun 5371

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 4131 df-opab 4193 df-rel 4781 df-cnv 4782 df-co 4783 df-fun 5379

This theorem is used by: funeq 5397 funopab4 5414 funres 5418 fun0 5439 funcnvcnv 5440 funin 5452 funres11 5453 foimacnv 5657 funsssuppss 6498 tfrlemibfn 6599 tfr1onlembfn 6615 tfrcllembfn 6628 strslssd 13399 strle1g 13460 subgrfun 16508