Theorem setsfun0 12810

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 12809 is useful for proofs based on isstruct2r 12785 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

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funres 5311	. . . . 5 |- ( Fun ( G \ { (/) } ) -> Fun ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) )
2	1	ad2antlr 489	. . . 4 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) )
3		funsng 5319	. . . . 5 |- ( ( I e. U /\ E e. W ) -> Fun { <. I , E >. } )
4	3	adantl 277	. . . 4 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun { <. I , E >. } )
5		dmres 4979	. . . . . . 7 |- dom ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) = ( ( _V \ dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) )
6	5	ineq1i 3369	. . . . . 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 >. } )
7		in32 3384	. . . . . . 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 \ { (/) } ) )
8		incom 3364	. . . . . . . . 9 |- ( ( _V \ dom { <. I , E >. } ) i^i dom { <. I , E >. } ) = ( dom { <. I , E >. } i^i ( _V \ dom { <. I , E >. } ) )
9		disjdif 3532	. . . . . . . . 9 |- ( dom { <. I , E >. } i^i ( _V \ dom { <. I , E >. } ) ) = (/)
10	8, 9	eqtri 2225	. . . . . . . 8 |- ( ( _V \ dom { <. I , E >. } ) i^i dom { <. I , E >. } ) = (/)
11	10	ineq1i 3369	. . . . . . 7 |- ( ( ( _V \ dom { <. I , E >. } ) i^i dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) ) = ( (/) i^i dom ( G \ { (/) } ) )
12		0in 3495	. . . . . . 7 |- ( (/) i^i dom ( G \ { (/) } ) ) = (/)
13	7, 11, 12	3eqtri 2229	. . . . . 6 |- ( ( ( _V \ dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) ) i^i dom { <. I , E >. } ) = (/)
14	6, 13	eqtri 2225	. . . . 5 |- ( dom ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) i^i dom { <. I , E >. } ) = (/)
15	14	a1i 9	. . . 4 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( dom ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) i^i dom { <. I , E >. } ) = (/) )
16		funun 5314	. . . 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 >. } ) )
17	2, 4, 15, 16	syl21anc 1248	. . 3 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
18		difundir 3425	. . . . 5 |- ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) \ { (/) } ) = ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) \ { (/) } ) u. ( { <. I , E >. } \ { (/) } ) )
19		resdifcom 4976	. . . . . . 7 |- ( ( G |` ( _V \ dom { <. I , E >. } ) ) \ { (/) } ) = ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) )
20	19	a1i 9	. . . . . 6 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( ( G |` ( _V \ dom { <. I , E >. } ) ) \ { (/) } ) = ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) )
21		elex 2782	. . . . . . . . 9 |- ( I e. U -> I e. _V )
22		elex 2782	. . . . . . . . 9 |- ( E e. W -> E e. _V )
23		opm 4277	. . . . . . . . . 10 |- ( E. x x e. <. I , E >. <-> ( I e. _V /\ E e. _V ) )
24		n0r 3473	. . . . . . . . . 10 |- ( E. x x e. <. I , E >. -> <. I , E >. =/= (/) )
25	23, 24	sylbir 135	. . . . . . . . 9 |- ( ( I e. _V /\ E e. _V ) -> <. I , E >. =/= (/) )
26	21, 22, 25	syl2an 289	. . . . . . . 8 |- ( ( I e. U /\ E e. W ) -> <. I , E >. =/= (/) )
27	26	adantl 277	. . . . . . 7 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> <. I , E >. =/= (/) )
28		disjsn2 3695	. . . . . . 7 |- ( <. I , E >. =/= (/) -> ( { <. I , E >. } i^i { (/) } ) = (/) )
29		disjdif2 3538	. . . . . . 7 |- ( ( { <. I , E >. } i^i { (/) } ) = (/) -> ( { <. I , E >. } \ { (/) } ) = { <. I , E >. } )
30	27, 28, 29	3syl 17	. . . . . 6 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( { <. I , E >. } \ { (/) } ) = { <. I , E >. } )
31	20, 30	uneq12d 3327	. . . . 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	18, 31	eqtrid 2249	. . . 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 5292	. . 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	17, 33	mpbird 167	. 2 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) \ { (/) } ) )
35		simpll 527	. . . . 5 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> G e. V )
36		opexg 4271	. . . . . 6 |- ( ( I e. U /\ E e. W ) -> <. I , E >. e. _V )
37	36	adantl 277	. . . . 5 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> <. I , E >. e. _V )
38		setsvalg 12804	. . . . 5 |- ( ( G e. V /\ <. I , E >. e. _V ) -> ( G sSet <. I , E >. ) = ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
39	35, 37, 38	syl2anc 411	. . . 4 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( G sSet <. I , E >. ) = ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
40	39	difeq1d 3289	. . 3 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> ( ( G sSet <. I , E >. ) \ { (/) } ) = ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) \ { (/) } ) )
41	40	funeqd 5292	. 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 167	1 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( G sSet <. I , E >. ) \ { (/) } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   E.wex 1514   e. wcel 2175   =/= wne 2375   _Vcvv 2771   \ cdif 3162   u. cun 3163   i^i cin 3164   (/)c0 3459   {csn 3632   <.cop 3635   dom cdm 4674   |` cres 4676   Fun wfun 5264  (class class class)co 5943   sSet csts 12772

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-res 4686  df-iota 5231  df-fun 5272  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-sets 12781

This theorem is referenced by:  setsn0fun  12811