Theorem gruima 9624

Description: A Grothendieck universe contains image sets drawn from its members. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
gruima	|- ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) -> ( A e. U -> ( F " A ) e. U ) )

Proof of Theorem gruima

Step	Hyp	Ref	Expression
1		simpl2 1065	. . . 4 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> Fun F )
2		funrel 5905	. . . 4 |- ( Fun F -> Rel F )
3		resres 5409	. . . . . . 7 |- ( ( F |` dom F ) |` A ) = ( F |` ( dom F i^i A ) )
4		resdm 5441	. . . . . . . 8 |- ( Rel F -> ( F |` dom F ) = F )
5	4	reseq1d 5395	. . . . . . 7 |- ( Rel F -> ( ( F |` dom F ) |` A ) = ( F |` A ) )
6	3, 5	syl5eqr 2670	. . . . . 6 |- ( Rel F -> ( F |` ( dom F i^i A ) ) = ( F |` A ) )
7	6	rneqd 5353	. . . . 5 |- ( Rel F -> ran ( F |` ( dom F i^i A ) ) = ran ( F |` A ) )
8		df-ima 5127	. . . . 5 |- ( F " A ) = ran ( F |` A )
9	7, 8	syl6reqr 2675	. . . 4 |- ( Rel F -> ( F " A ) = ran ( F |` ( dom F i^i A ) ) )
10	1, 2, 9	3syl 18	. . 3 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> ( F " A ) = ran ( F |` ( dom F i^i A ) ) )
11		simpl1 1064	. . . 4 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> U e. Univ )
12		simpr 477	. . . . 5 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> A e. U )
13		inss2 3834	. . . . . 6 |- ( dom F i^i A ) C_ A
14	13	a1i 11	. . . . 5 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> ( dom F i^i A ) C_ A )
15		gruss 9618	. . . . 5 |- ( ( U e. Univ /\ A e. U /\ ( dom F i^i A ) C_ A ) -> ( dom F i^i A ) e. U )
16	11, 12, 14, 15	syl3anc 1326	. . . 4 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> ( dom F i^i A ) e. U )
17		funforn 6122	. . . . . . . 8 |- ( Fun F <-> F : dom F -onto-> ran F )
18		fof 6115	. . . . . . . 8 |- ( F : dom F -onto-> ran F -> F : dom F --> ran F )
19	17, 18	sylbi 207	. . . . . . 7 |- ( Fun F -> F : dom F --> ran F )
20		inss1 3833	. . . . . . 7 |- ( dom F i^i A ) C_ dom F
21		fssres 6070	. . . . . . 7 |- ( ( F : dom F --> ran F /\ ( dom F i^i A ) C_ dom F ) -> ( F |` ( dom F i^i A ) ) : ( dom F i^i A ) --> ran F )
22	19, 20, 21	sylancl 694	. . . . . 6 |- ( Fun F -> ( F |` ( dom F i^i A ) ) : ( dom F i^i A ) --> ran F )
23		ffn 6045	. . . . . 6 |- ( ( F |` ( dom F i^i A ) ) : ( dom F i^i A ) --> ran F -> ( F |` ( dom F i^i A ) ) Fn ( dom F i^i A ) )
24	1, 22, 23	3syl 18	. . . . 5 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> ( F |` ( dom F i^i A ) ) Fn ( dom F i^i A ) )
25		simpl3 1066	. . . . . 6 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> ( F " A ) C_ U )
26	10, 25	eqsstr3d 3640	. . . . 5 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> ran ( F |` ( dom F i^i A ) ) C_ U )
27		df-f 5892	. . . . 5 |- ( ( F |` ( dom F i^i A ) ) : ( dom F i^i A ) --> U <-> ( ( F |` ( dom F i^i A ) ) Fn ( dom F i^i A ) /\ ran ( F |` ( dom F i^i A ) ) C_ U ) )
28	24, 26, 27	sylanbrc 698	. . . 4 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> ( F |` ( dom F i^i A ) ) : ( dom F i^i A ) --> U )
29		grurn 9623	. . . 4 |- ( ( U e. Univ /\ ( dom F i^i A ) e. U /\ ( F |` ( dom F i^i A ) ) : ( dom F i^i A ) --> U ) -> ran ( F |` ( dom F i^i A ) ) e. U )
30	11, 16, 28, 29	syl3anc 1326	. . 3 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> ran ( F |` ( dom F i^i A ) ) e. U )
31	10, 30	eqeltrd 2701	. 2 |- ( ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) /\ A e. U ) -> ( F " A ) e. U )
32	31	ex 450	1 |- ( ( U e. Univ /\ Fun F /\ ( F " A ) C_ U ) -> ( A e. U -> ( F " A ) e. U ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   Univcgru 9612

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-pow 4843  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-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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-gru 9613

This theorem is referenced by: (None)