Theorem setsn0fun 13109

Description: The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set) is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)

Hypotheses

Ref	Expression
setsn0fun.s	|- ( ph -> S Struct X )
setsn0fun.i	|- ( ph -> I e. U )
setsn0fun.e	|- ( ph -> E e. W )

Assertion

Ref	Expression
setsn0fun	|- ( ph -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) )

Proof of Theorem setsn0fun

Step	Hyp	Ref	Expression
1		setsn0fun.s	. 2 |- ( ph -> S Struct X )
2		structn0fun 13085	. . 3 |- ( S Struct X -> Fun ( S \ { (/) } ) )
3		setsn0fun.i	. . . . 5 |- ( ph -> I e. U )
4		setsn0fun.e	. . . . 5 |- ( ph -> E e. W )
5		structex 13084	. . . . . . 7 |- ( S Struct X -> S e. _V )
6		setsfun0 13108	. . . . . . 7 |- ( ( ( S e. _V /\ Fun ( S \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) )
7	5, 6	sylanl1 402	. . . . . 6 |- ( ( ( S Struct X /\ Fun ( S \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) )
8	7	expcom 116	. . . . 5 |- ( ( I e. U /\ E e. W ) -> ( ( S Struct X /\ Fun ( S \ { (/) } ) ) -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) ) )
9	3, 4, 8	syl2anc 411	. . . 4 |- ( ph -> ( ( S Struct X /\ Fun ( S \ { (/) } ) ) -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) ) )
10	9	com12 30	. . 3 |- ( ( S Struct X /\ Fun ( S \ { (/) } ) ) -> ( ph -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) ) )
11	2, 10	mpdan 421	. 2 |- ( S Struct X -> ( ph -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) ) )
12	1, 11	mpcom 36	1 |- ( ph -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   _Vcvv 2800   \ cdif 3195   (/)c0 3492   {csn 3667   <.cop 3670   class class class wbr 4086   Fun wfun 5318  (class class class)co 6013   Struct cstr 13068   sSet csts 13070

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-res 4735  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-struct 13074  df-sets 13079

This theorem is referenced by:  setsvtx  15892  setsiedg  15893