Theorem setsn0fun 12655

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 12631	. . 3 |- ( S Struct X -> Fun ( S \ { (/) } ) )
3		setsn0fun.i	. . . . 5 |- ( ph -> I e. U )
4		setsn0fun.e	. . . . 5 |- ( ph -> E e. W )
5		structex 12630	. . . . . . 7 |- ( S Struct X -> S e. _V )
6		setsfun0 12654	. . . . . . 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 2164   _Vcvv 2760   \ cdif 3150   (/)c0 3446   {csn 3618   <.cop 3621   class class class wbr 4029   Fun wfun 5248  (class class class)co 5918   Struct cstr 12614   sSet csts 12616

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-res 4671  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-struct 12620  df-sets 12625

This theorem is referenced by: (None)