Theorem setsn0fun 13238

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 13214	. . 3 |- ( S Struct X -> Fun ( S \ { (/) } ) )
3		setsn0fun.i	. . . . 5 |- ( ph -> I e. U )
4		setsn0fun.e	. . . . 5 |- ( ph -> E e. W )
5		structex 13213	. . . . . . 7 |- ( S Struct X -> S e. _V )
6		setsfun0 13237	. . . . . . 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 2203   _Vcvv 2812   \ cdif 3207   (/)c0 3507   {csn 3688   <.cop 3691   class class class wbr 4108   Fun wfun 5345  (class class class)co 6049   Struct cstr 13197   sSet csts 13199

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-res 4760  df-iota 5311  df-fun 5353  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-struct 13203  df-sets 13208

This theorem is referenced by:  setsvtx  16033  setsiedg  16034