Theorem setsfun0 15894

Description: A structure with replacement without the empty set is a function if the original structure without the empty set is a function. This variant of setsfun 15893 is useful for proofs based on isstruct2 15867 which requires Fun ( F \ { (/) } ) for F to be an extensible structure. (Contributed by AV, 7-Jun-2021.)

Assertion

Ref	Expression
setsfun0	|- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( G sSet <. I , E >. ) \ { (/) } ) )

Proof of Theorem setsfun0

Step	Hyp	Ref	Expression
1		funres 5929	. . . . . 6 |- ( Fun ( G \ { (/) } ) -> Fun ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) )
2	1	adantl 482	. . . . 5 |- ( ( G e. V /\ Fun ( G \ { (/) } ) ) -> Fun ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) )
3	2	adantr 481	. . . 4 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) )
4		funsng 5937	. . . . 5 |- ( ( I e. U /\ E e. W ) -> Fun { <. I , E >. } )
5	4	adantl 482	. . . 4 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun { <. I , E >. } )
6		dmres 5419	. . . . . . 7 |- dom ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) = ( ( _V \ dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) )
7	6	ineq1i 3810	. . . . . 6 |- ( dom ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) i^i dom { <. I , E >. } ) = ( ( ( _V \ dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) ) i^i dom { <. I , E >. } )
8		in32 3825	. . . . . . 7 |- ( ( ( _V \ dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) ) i^i dom { <. I , E >. } ) = ( ( ( _V \ dom { <. I , E >. } ) i^i dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) )
9		incom 3805	. . . . . . . . 9 |- ( ( _V \ dom { <. I , E >. } ) i^i dom { <. I , E >. } ) = ( dom { <. I , E >. } i^i ( _V \ dom { <. I , E >. } ) )
10		disjdif 4040	. . . . . . . . 9 |- ( dom { <. I , E >. } i^i ( _V \ dom { <. I , E >. } ) ) = (/)
11	9, 10	eqtri 2644	. . . . . . . 8 |- ( ( _V \ dom { <. I , E >. } ) i^i dom { <. I , E >. } ) = (/)
12	11	ineq1i 3810	. . . . . . 7 |- ( ( ( _V \ dom { <. I , E >. } ) i^i dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) ) = ( (/) i^i dom ( G \ { (/) } ) )
13		0in 3969	. . . . . . 7 |- ( (/) i^i dom ( G \ { (/) } ) ) = (/)
14	8, 12, 13	3eqtri 2648	. . . . . 6 |- ( ( ( _V \ dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) ) i^i dom { <. I , E >. } ) = (/)
15	7, 14	eqtri 2644	. . . . 5 |- ( dom ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) i^i dom { <. I , E >. } ) = (/)
16	15	a1i 11	. . . 4 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( dom ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) i^i dom { <. I , E >. } ) = (/) )
17		funun 5932	. . . 4 |- ( ( ( Fun ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) /\ Fun { <. I , E >. } ) /\ ( dom ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) i^i dom { <. I , E >. } ) = (/) ) -> Fun ( ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
18	3, 5, 16, 17	syl21anc 1325	. . 3 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
19		difundir 3880	. . . . 5 |- ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) \ { (/) } ) = ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) \ { (/) } ) u. ( { <. I , E >. } \ { (/) } ) )
20		resdifcom 5415	. . . . . . 7 |- ( ( G |` ( _V \ dom { <. I , E >. } ) ) \ { (/) } ) = ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) )
21	20	a1i 11	. . . . . 6 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( ( G |` ( _V \ dom { <. I , E >. } ) ) \ { (/) } ) = ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) )
22		elex 3212	. . . . . . . . . 10 |- ( I e. U -> I e. _V )
23		elex 3212	. . . . . . . . . 10 |- ( E e. W -> E e. _V )
24	22, 23	anim12i 590	. . . . . . . . 9 |- ( ( I e. U /\ E e. W ) -> ( I e. _V /\ E e. _V ) )
25		opnz 4942	. . . . . . . . 9 |- ( <. I , E >. =/= (/) <-> ( I e. _V /\ E e. _V ) )
26	24, 25	sylibr 224	. . . . . . . 8 |- ( ( I e. U /\ E e. W ) -> <. I , E >. =/= (/) )
27	26	adantl 482	. . . . . . 7 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> <. I , E >. =/= (/) )
28		disjsn2 4247	. . . . . . 7 |- ( <. I , E >. =/= (/) -> ( { <. I , E >. } i^i { (/) } ) = (/) )
29		disjdif2 4047	. . . . . . 7 |- ( ( { <. I , E >. } i^i { (/) } ) = (/) -> ( { <. I , E >. } \ { (/) } ) = { <. I , E >. } )
30	27, 28, 29	3syl 18	. . . . . 6 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( { <. I , E >. } \ { (/) } ) = { <. I , E >. } )
31	21, 30	uneq12d 3768	. . . . 5 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) \ { (/) } ) u. ( { <. I , E >. } \ { (/) } ) ) = ( ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
32	19, 31	syl5eq 2668	. . . 4 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) \ { (/) } ) = ( ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
33	32	funeqd 5910	. . 3 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( Fun ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) \ { (/) } ) <-> Fun ( ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) ) )
34	18, 33	mpbird 247	. 2 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) \ { (/) } ) )
35		opex 4932	. . . . . . 7 |- <. I , E >. e. _V
36	35	a1i 11	. . . . . 6 |- ( Fun ( G \ { (/) } ) -> <. I , E >. e. _V )
37		setsvalg 15887	. . . . . 6 |- ( ( G e. V /\ <. I , E >. e. _V ) -> ( G sSet <. I , E >. ) = ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
38	36, 37	sylan2 491	. . . . 5 |- ( ( G e. V /\ Fun ( G \ { (/) } ) ) -> ( G sSet <. I , E >. ) = ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
39	38	difeq1d 3727	. . . 4 |- ( ( G e. V /\ Fun ( G \ { (/) } ) ) -> ( ( G sSet <. I , E >. ) \ { (/) } ) = ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) \ { (/) } ) )
40	39	funeqd 5910	. . 3 |- ( ( G e. V /\ Fun ( G \ { (/) } ) ) -> ( Fun ( ( G sSet <. I , E >. ) \ { (/) } ) <-> Fun ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) \ { (/) } ) ) )
41	40	adantr 481	. 2 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( Fun ( ( G sSet <. I , E >. ) \ { (/) } ) <-> Fun ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) \ { (/) } ) ) )
42	34, 41	mpbird 247	1 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( G sSet <. I , E >. ) \ { (/) } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   <.cop 4183   dom cdm 5114   |` cres 5116   Fun wfun 5882  (class class class)co 6650   sSet csts 15855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sets 15864

This theorem is referenced by:  setsn0fun  15895  setsstruct2  15896  setsstructOLD  15899