Theorem wunfunc 16559

Description: A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
wunfunc.1	|- ( ph -> U e. WUni )
wunfunc.2	|- ( ph -> C e. U )
wunfunc.3	|- ( ph -> D e. U )

Assertion

Ref	Expression
wunfunc	|- ( ph -> ( C Func D ) e. U )

Proof of Theorem wunfunc

Dummy variables f g z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wunfunc.1	. 2 |- ( ph -> U e. WUni )
2		df-base 15863	. . . . 5 |- Base = Slot 1
3		wunfunc.3	. . . . 5 |- ( ph -> D e. U )
4	2, 1, 3	wunstr 15881	. . . 4 |- ( ph -> ( Base ` D ) e. U )
5		wunfunc.2	. . . . 5 |- ( ph -> C e. U )
6	2, 1, 5	wunstr 15881	. . . 4 |- ( ph -> ( Base ` C ) e. U )
7	1, 4, 6	wunmap 9548	. . 3 |- ( ph -> ( ( Base ` D ) ^m ( Base ` C ) ) e. U )
8		df-hom 15966	. . . . . . . . 9 |- Hom = Slot ; 1 4
9	8, 1, 5	wunstr 15881	. . . . . . . 8 |- ( ph -> ( Hom ` C ) e. U )
10	1, 9	wunrn 9551	. . . . . . 7 |- ( ph -> ran ( Hom ` C ) e. U )
11	1, 10	wununi 9528	. . . . . 6 |- ( ph -> U. ran ( Hom ` C ) e. U )
12	8, 1, 3	wunstr 15881	. . . . . . . 8 |- ( ph -> ( Hom ` D ) e. U )
13	1, 12	wunrn 9551	. . . . . . 7 |- ( ph -> ran ( Hom ` D ) e. U )
14	1, 13	wununi 9528	. . . . . 6 |- ( ph -> U. ran ( Hom ` D ) e. U )
15	1, 11, 14	wunxp 9546	. . . . 5 |- ( ph -> ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) e. U )
16	1, 15	wunpw 9529	. . . 4 |- ( ph -> ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) e. U )
17	1, 6, 6	wunxp 9546	. . . 4 |- ( ph -> ( ( Base ` C ) X. ( Base ` C ) ) e. U )
18	1, 16, 17	wunmap 9548	. . 3 |- ( ph -> ( ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) ^m ( ( Base ` C ) X. ( Base ` C ) ) ) e. U )
19	1, 7, 18	wunxp 9546	. 2 |- ( ph -> ( ( ( Base ` D ) ^m ( Base ` C ) ) X. ( ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) ^m ( ( Base ` C ) X. ( Base ` C ) ) ) ) e. U )
20		relfunc 16522	. . . 4 |- Rel ( C Func D )
21	20	a1i 11	. . 3 |- ( ph -> Rel ( C Func D ) )
22		df-br 4654	. . . 4 |- ( f ( C Func D ) g <-> <. f , g >. e. ( C Func D ) )
23		eqid 2622	. . . . . . . 8 |- ( Base ` C ) = ( Base ` C )
24		eqid 2622	. . . . . . . 8 |- ( Base ` D ) = ( Base ` D )
25		simpr 477	. . . . . . . 8 |- ( ( ph /\ f ( C Func D ) g ) -> f ( C Func D ) g )
26	23, 24, 25	funcf1 16526	. . . . . . 7 |- ( ( ph /\ f ( C Func D ) g ) -> f : ( Base ` C ) --> ( Base ` D ) )
27		fvex 6201	. . . . . . . 8 |- ( Base ` D ) e. _V
28		fvex 6201	. . . . . . . 8 |- ( Base ` C ) e. _V
29	27, 28	elmap 7886	. . . . . . 7 |- ( f e. ( ( Base ` D ) ^m ( Base ` C ) ) <-> f : ( Base ` C ) --> ( Base ` D ) )
30	26, 29	sylibr 224	. . . . . 6 |- ( ( ph /\ f ( C Func D ) g ) -> f e. ( ( Base ` D ) ^m ( Base ` C ) ) )
31		mapsspw 7893	. . . . . . . . . . 11 |- ( ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ^m ( ( Hom ` C ) ` z ) ) C_ ~P ( ( ( Hom ` C ) ` z ) X. ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) )
32		fvssunirn 6217	. . . . . . . . . . . . 13 |- ( ( Hom ` C ) ` z ) C_ U. ran ( Hom ` C )
33		ovssunirn 6681	. . . . . . . . . . . . 13 |- ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) C_ U. ran ( Hom ` D )
34		xpss12 5225	. . . . . . . . . . . . 13 |- ( ( ( ( Hom ` C ) ` z ) C_ U. ran ( Hom ` C ) /\ ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) C_ U. ran ( Hom ` D ) ) -> ( ( ( Hom ` C ) ` z ) X. ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ) C_ ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) )
35	32, 33, 34	mp2an 708	. . . . . . . . . . . 12 |- ( ( ( Hom ` C ) ` z ) X. ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ) C_ ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) )
36		sspwb 4917	. . . . . . . . . . . 12 |- ( ( ( ( Hom ` C ) ` z ) X. ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ) C_ ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) <-> ~P ( ( ( Hom ` C ) ` z ) X. ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ) C_ ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) )
37	35, 36	mpbi 220	. . . . . . . . . . 11 |- ~P ( ( ( Hom ` C ) ` z ) X. ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ) C_ ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) )
38	31, 37	sstri 3612	. . . . . . . . . 10 |- ( ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ^m ( ( Hom ` C ) ` z ) ) C_ ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) )
39	38	rgenw 2924	. . . . . . . . 9 |- A. z e. ( ( Base ` C ) X. ( Base ` C ) ) ( ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ^m ( ( Hom ` C ) ` z ) ) C_ ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) )
40		ss2ixp 7921	. . . . . . . . 9 |- ( A. z e. ( ( Base ` C ) X. ( Base ` C ) ) ( ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ^m ( ( Hom ` C ) ` z ) ) C_ ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) -> X_ z e. ( ( Base ` C ) X. ( Base ` C ) ) ( ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ^m ( ( Hom ` C ) ` z ) ) C_ X_ z e. ( ( Base ` C ) X. ( Base ` C ) ) ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) )
41	39, 40	ax-mp 5	. . . . . . . 8 |- X_ z e. ( ( Base ` C ) X. ( Base ` C ) ) ( ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ^m ( ( Hom ` C ) ` z ) ) C_ X_ z e. ( ( Base ` C ) X. ( Base ` C ) ) ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) )
42	28, 28	xpex 6962	. . . . . . . . 9 |- ( ( Base ` C ) X. ( Base ` C ) ) e. _V
43		fvex 6201	. . . . . . . . . . . . 13 |- ( Hom ` C ) e. _V
44	43	rnex 7100	. . . . . . . . . . . 12 |- ran ( Hom ` C ) e. _V
45	44	uniex 6953	. . . . . . . . . . 11 |- U. ran ( Hom ` C ) e. _V
46		fvex 6201	. . . . . . . . . . . . 13 |- ( Hom ` D ) e. _V
47	46	rnex 7100	. . . . . . . . . . . 12 |- ran ( Hom ` D ) e. _V
48	47	uniex 6953	. . . . . . . . . . 11 |- U. ran ( Hom ` D ) e. _V
49	45, 48	xpex 6962	. . . . . . . . . 10 |- ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) e. _V
50	49	pwex 4848	. . . . . . . . 9 |- ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) e. _V
51	42, 50	ixpconst 7918	. . . . . . . 8 |- X_ z e. ( ( Base ` C ) X. ( Base ` C ) ) ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) = ( ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) ^m ( ( Base ` C ) X. ( Base ` C ) ) )
52	41, 51	sseqtri 3637	. . . . . . 7 |- X_ z e. ( ( Base ` C ) X. ( Base ` C ) ) ( ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ^m ( ( Hom ` C ) ` z ) ) C_ ( ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) ^m ( ( Base ` C ) X. ( Base ` C ) ) )
53		eqid 2622	. . . . . . . 8 |- ( Hom ` C ) = ( Hom ` C )
54		eqid 2622	. . . . . . . 8 |- ( Hom ` D ) = ( Hom ` D )
55	23, 53, 54, 25	funcixp 16527	. . . . . . 7 |- ( ( ph /\ f ( C Func D ) g ) -> g e. X_ z e. ( ( Base ` C ) X. ( Base ` C ) ) ( ( ( f ` ( 1st ` z ) ) ( Hom ` D ) ( f ` ( 2nd ` z ) ) ) ^m ( ( Hom ` C ) ` z ) ) )
56	52, 55	sseldi 3601	. . . . . 6 |- ( ( ph /\ f ( C Func D ) g ) -> g e. ( ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) ^m ( ( Base ` C ) X. ( Base ` C ) ) ) )
57		opelxpi 5148	. . . . . 6 |- ( ( f e. ( ( Base ` D ) ^m ( Base ` C ) ) /\ g e. ( ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) ^m ( ( Base ` C ) X. ( Base ` C ) ) ) ) -> <. f , g >. e. ( ( ( Base ` D ) ^m ( Base ` C ) ) X. ( ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) ^m ( ( Base ` C ) X. ( Base ` C ) ) ) ) )
58	30, 56, 57	syl2anc 693	. . . . 5 |- ( ( ph /\ f ( C Func D ) g ) -> <. f , g >. e. ( ( ( Base ` D ) ^m ( Base ` C ) ) X. ( ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) ^m ( ( Base ` C ) X. ( Base ` C ) ) ) ) )
59	58	ex 450	. . . 4 |- ( ph -> ( f ( C Func D ) g -> <. f , g >. e. ( ( ( Base ` D ) ^m ( Base ` C ) ) X. ( ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) ^m ( ( Base ` C ) X. ( Base ` C ) ) ) ) ) )
60	22, 59	syl5bir 233	. . 3 |- ( ph -> ( <. f , g >. e. ( C Func D ) -> <. f , g >. e. ( ( ( Base ` D ) ^m ( Base ` C ) ) X. ( ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) ^m ( ( Base ` C ) X. ( Base ` C ) ) ) ) ) )
61	21, 60	relssdv 5212	. 2 |- ( ph -> ( C Func D ) C_ ( ( ( Base ` D ) ^m ( Base ` C ) ) X. ( ~P ( U. ran ( Hom ` C ) X. U. ran ( Hom ` D ) ) ^m ( ( Base ` C ) X. ( Base ` C ) ) ) ) )
62	1, 19, 61	wunss 9534	1 |- ( ph -> ( C Func D ) e. U )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   C_ wss 3574   ~Pcpw 4158   <.cop 4183   U.cuni 4436   class class class wbr 4653   X. cxp 5112   ran crn 5115   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   X_cixp 7908  WUnicwun 9522   1c1 9937   4c4 11072  ;cdc 11493   Basecbs 15857   Hom chom 15952   Func cfunc 16514

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-rep 4771  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-pm 7860  df-ixp 7909  df-wun 9524  df-slot 15861  df-base 15863  df-hom 15966  df-func 16518

This theorem is referenced by:  wunnat  16616  catcfuccl  16759