Theorem setsn0fun 13120

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 13096	. . 3 |- ( S Struct X -> Fun ( S \ { (/) } ) )
3		setsn0fun.i	. . . . 5 |- ( ph -> I e. U )
4		setsn0fun.e	. . . . 5 |- ( ph -> E e. W )
5		structex 13095	. . . . . . 7 |- ( S Struct X -> S e. _V )
6		setsfun0 13119	. . . . . . 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 2202   _Vcvv 2802   \ cdif 3197   (/)c0 3494   {csn 3669   <.cop 3672   class class class wbr 4088   Fun wfun 5320  (class class class)co 6018   Struct cstr 13079   sSet csts 13081

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-struct 13085  df-sets 13090

This theorem is referenced by:  setsvtx  15904  setsiedg  15905