Theorem unirnmap 39400

Description: Given a subset of a set exponentiation, the base set can be restricted. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
unirnmap.a	|- ( ph -> A e. V )
unirnmap.x	|- ( ph -> X C_ ( B ^m A ) )

Assertion

Ref	Expression
unirnmap	|- ( ph -> X C_ ( ran U. X ^m A ) )

Proof of Theorem unirnmap

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

Step	Hyp	Ref	Expression
1		unirnmap.x	. . . . . . . 8 |- ( ph -> X C_ ( B ^m A ) )
2	1	sselda 3603	. . . . . . 7 |- ( ( ph /\ g e. X ) -> g e. ( B ^m A ) )
3		elmapfn 7880	. . . . . . 7 |- ( g e. ( B ^m A ) -> g Fn A )
4	2, 3	syl 17	. . . . . 6 |- ( ( ph /\ g e. X ) -> g Fn A )
5		simplr 792	. . . . . . . . . 10 |- ( ( ( ph /\ g e. X ) /\ x e. A ) -> g e. X )
6		dffn3 6054	. . . . . . . . . . . 12 |- ( g Fn A <-> g : A --> ran g )
7	4, 6	sylib 208	. . . . . . . . . . 11 |- ( ( ph /\ g e. X ) -> g : A --> ran g )
8	7	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ( ph /\ g e. X ) /\ x e. A ) -> ( g ` x ) e. ran g )
9		rneq 5351	. . . . . . . . . . . 12 |- ( f = g -> ran f = ran g )
10	9	eleq2d 2687	. . . . . . . . . . 11 |- ( f = g -> ( ( g ` x ) e. ran f <-> ( g ` x ) e. ran g ) )
11	10	rspcev 3309	. . . . . . . . . 10 |- ( ( g e. X /\ ( g ` x ) e. ran g ) -> E. f e. X ( g ` x ) e. ran f )
12	5, 8, 11	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ g e. X ) /\ x e. A ) -> E. f e. X ( g ` x ) e. ran f )
13		eliun 4524	. . . . . . . . 9 |- ( ( g ` x ) e. U_ f e. X ran f <-> E. f e. X ( g ` x ) e. ran f )
14	12, 13	sylibr 224	. . . . . . . 8 |- ( ( ( ph /\ g e. X ) /\ x e. A ) -> ( g ` x ) e. U_ f e. X ran f )
15		rnuni 5544	. . . . . . . 8 |- ran U. X = U_ f e. X ran f
16	14, 15	syl6eleqr 2712	. . . . . . 7 |- ( ( ( ph /\ g e. X ) /\ x e. A ) -> ( g ` x ) e. ran U. X )
17	16	ralrimiva 2966	. . . . . 6 |- ( ( ph /\ g e. X ) -> A. x e. A ( g ` x ) e. ran U. X )
18	4, 17	jca 554	. . . . 5 |- ( ( ph /\ g e. X ) -> ( g Fn A /\ A. x e. A ( g ` x ) e. ran U. X ) )
19		ffnfv 6388	. . . . 5 |- ( g : A --> ran U. X <-> ( g Fn A /\ A. x e. A ( g ` x ) e. ran U. X ) )
20	18, 19	sylibr 224	. . . 4 |- ( ( ph /\ g e. X ) -> g : A --> ran U. X )
21		ovexd 6680	. . . . . . . . 9 |- ( ph -> ( B ^m A ) e. _V )
22	21, 1	ssexd 4805	. . . . . . . 8 |- ( ph -> X e. _V )
23		uniexg 6955	. . . . . . . 8 |- ( X e. _V -> U. X e. _V )
24	22, 23	syl 17	. . . . . . 7 |- ( ph -> U. X e. _V )
25		rnexg 7098	. . . . . . 7 |- ( U. X e. _V -> ran U. X e. _V )
26	24, 25	syl 17	. . . . . 6 |- ( ph -> ran U. X e. _V )
27		unirnmap.a	. . . . . 6 |- ( ph -> A e. V )
28	26, 27	elmapd 7871	. . . . 5 |- ( ph -> ( g e. ( ran U. X ^m A ) <-> g : A --> ran U. X ) )
29	28	adantr 481	. . . 4 |- ( ( ph /\ g e. X ) -> ( g e. ( ran U. X ^m A ) <-> g : A --> ran U. X ) )
30	20, 29	mpbird 247	. . 3 |- ( ( ph /\ g e. X ) -> g e. ( ran U. X ^m A ) )
31	30	ralrimiva 2966	. 2 |- ( ph -> A. g e. X g e. ( ran U. X ^m A ) )
32		dfss3 3592	. 2 |- ( X C_ ( ran U. X ^m A ) <-> A. g e. X g e. ( ran U. X ^m A ) )
33	31, 32	sylibr 224	1 |- ( ph -> X C_ ( ran U. X ^m A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   U.cuni 4436   U_ciun 4520   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857

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-ne 2795  df-ral 2917  df-rex 2918  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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  unirnmapsn  39406