Theorem setsfun0 13248

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 13247 is useful for proofs based on isstruct2r 13223 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 5393	. . . . 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 5402	. . . . 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 5059	. . . . . . 7 |- dom ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) = ( ( _V \ dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) )
6	5	ineq1i 3418	. . . . . 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 3433	. . . . . . 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 3411	. . . . . . . . 9 |- ( ( _V \ dom { <. I , E >. } ) i^i dom { <. I , E >. } ) = ( dom { <. I , E >. } i^i ( _V \ dom { <. I , E >. } ) )
9		disjdif 3581	. . . . . . . . 9 |- ( dom { <. I , E >. } i^i ( _V \ dom { <. I , E >. } ) ) = (/)
10	8, 9	eqtri 2253	. . . . . . . 8 |- ( ( _V \ dom { <. I , E >. } ) i^i dom { <. I , E >. } ) = (/)
11	10	ineq1i 3418	. . . . . . 7 |- ( ( ( _V \ dom { <. I , E >. } ) i^i dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) ) = ( (/) i^i dom ( G \ { (/) } ) )
12		0in 3544	. . . . . . 7 |- ( (/) i^i dom ( G \ { (/) } ) ) = (/)
13	7, 11, 12	3eqtri 2257	. . . . . 6 |- ( ( ( _V \ dom { <. I , E >. } ) i^i dom ( G \ { (/) } ) ) i^i dom { <. I , E >. } ) = (/)
14	6, 13	eqtri 2253	. . . . 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 5397	. . . 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 1273	. . 3 |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( ( G \ { (/) } ) |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) )
18		difundir 3474	. . . . 5 |- ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) u. { <. I , E >. } ) \ { (/) } ) = ( ( ( G |` ( _V \ dom { <. I , E >. } ) ) \ { (/) } ) u. ( { <. I , E >. } \ { (/) } ) )
19		resdifcom 5056	. . . . . . 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 2825	. . . . . . . . 9 |- ( I e. U -> I e. _V )
22		elex 2825	. . . . . . . . 9 |- ( E e. W -> E e. _V )
23		opm 4350	. . . . . . . . . 10 |- ( E. x x e. <. I , E >. <-> ( I e. _V /\ E e. _V ) )
24		n0r 3522	. . . . . . . . . 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 3752	. . . . . . 7 |- ( <. I , E >. =/= (/) -> ( { <. I , E >. } i^i { (/) } ) = (/) )
29		disjdif2 3588	. . . . . . 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 3374	. . . . 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 2277	. . . 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 5374	. . 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 4344	. . . . . 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 13242	. . . . 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 3336	. . 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 5374	. 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 1398   E.wex 1541   e. wcel 2203   =/= wne 2412   _Vcvv 2813   \ cdif 3208   u. cun 3209   i^i cin 3210   (/)c0 3508   {csn 3689   <.cop 3692   dom cdm 4749   |` cres 4751   Fun wfun 5346  (class class class)co 6050   sSet csts 13210

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 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

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 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sets 13219

This theorem is referenced by:  setsn0fun  13249