Theorem subgrfun 16422

Description: The edge function of a subgraph of a graph whose edge function is actually a function is a function. (Contributed by AV, 20-Nov-2020.)

Assertion

Ref	Expression
subgrfun	|- ( ( Fun (iEdg ` G ) /\ S SubGraph G ) -> Fun (iEdg ` S ) )

Proof of Theorem subgrfun

Step	Hyp	Ref	Expression
1		eqid 2238	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
2		eqid 2238	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2238	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
4		eqid 2238	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
5		eqid 2238	. . . 4 |- (Edg ` S ) = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 16415	. . 3 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
7		funss 5391	. . . 4 |- ( (iEdg ` S ) C_ (iEdg ` G ) -> ( Fun (iEdg ` G ) -> Fun (iEdg ` S ) ) )
8	7	3ad2ant2 1050	. . 3 |- ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) -> ( Fun (iEdg ` G ) -> Fun (iEdg ` S ) ) )
9	6, 8	syl 14	. 2 |- ( S SubGraph G -> ( Fun (iEdg ` G ) -> Fun (iEdg ` S ) ) )
10	9	impcom 125	1 |- ( ( Fun (iEdg ` G ) /\ S SubGraph G ) -> Fun (iEdg ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   Fun wfun 5366   ` cfv 5372  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212   SubGraph csubgr 16408

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 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-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-res 4781  df-iota 5332  df-fun 5374  df-fv 5380  df-subgr 16409

This theorem is referenced by:  subgruhgrfun  16423