Theorem madeval 31935

Description: The value of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)

Assertion

Ref	Expression
madeval	|- ( A e. On -> ( M ` A ) = ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) )

Proof of Theorem madeval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-made 31930	. . 3 |- M = recs ( ( x e. _V |-> ( |s " ( ~P U. ran x X. ~P U. ran x ) ) ) )
2	1	tfr2 7494	. 2 |- ( A e. On -> ( M ` A ) = ( ( x e. _V |-> ( |s " ( ~P U. ran x X. ~P U. ran x ) ) ) ` ( M |` A ) ) )
3	1	tfr1 7493	. . . . 5 |- M Fn On
4		fnfun 5988	. . . . 5 |- ( M Fn On -> Fun M )
5	3, 4	ax-mp 5	. . . 4 |- Fun M
6		resfunexg 6479	. . . 4 |- ( ( Fun M /\ A e. On ) -> ( M |` A ) e. _V )
7	5, 6	mpan 706	. . 3 |- ( A e. On -> ( M |` A ) e. _V )
8		scutf 31919	. . . . 5 |- |s : < <s --> No
9		ffun 6048	. . . . 5 |- ( |s : < <s --> No -> Fun |s )
10	8, 9	ax-mp 5	. . . 4 |- Fun |s
11		funimaexg 5975	. . . . . . 7 |- ( ( Fun M /\ A e. On ) -> ( M " A ) e. _V )
12	5, 11	mpan 706	. . . . . 6 |- ( A e. On -> ( M " A ) e. _V )
13		uniexg 6955	. . . . . 6 |- ( ( M " A ) e. _V -> U. ( M " A ) e. _V )
14		pwexg 4850	. . . . . 6 |- ( U. ( M " A ) e. _V -> ~P U. ( M " A ) e. _V )
15	12, 13, 14	3syl 18	. . . . 5 |- ( A e. On -> ~P U. ( M " A ) e. _V )
16		xpexg 6960	. . . . 5 |- ( ( ~P U. ( M " A ) e. _V /\ ~P U. ( M " A ) e. _V ) -> ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) e. _V )
17	15, 15, 16	syl2anc 693	. . . 4 |- ( A e. On -> ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) e. _V )
18		funimaexg 5975	. . . 4 |- ( ( Fun |s /\ ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) e. _V ) -> ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) e. _V )
19	10, 17, 18	sylancr 695	. . 3 |- ( A e. On -> ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) e. _V )
20		rneq 5351	. . . . . . . . 9 |- ( x = ( M |` A ) -> ran x = ran ( M |` A ) )
21		df-ima 5127	. . . . . . . . 9 |- ( M " A ) = ran ( M |` A )
22	20, 21	syl6eqr 2674	. . . . . . . 8 |- ( x = ( M |` A ) -> ran x = ( M " A ) )
23	22	unieqd 4446	. . . . . . 7 |- ( x = ( M |` A ) -> U. ran x = U. ( M " A ) )
24	23	pweqd 4163	. . . . . 6 |- ( x = ( M |` A ) -> ~P U. ran x = ~P U. ( M " A ) )
25	24	sqxpeqd 5141	. . . . 5 |- ( x = ( M |` A ) -> ( ~P U. ran x X. ~P U. ran x ) = ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) )
26	25	imaeq2d 5466	. . . 4 |- ( x = ( M |` A ) -> ( |s " ( ~P U. ran x X. ~P U. ran x ) ) = ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) )
27		eqid 2622	. . . 4 |- ( x e. _V |-> ( |s " ( ~P U. ran x X. ~P U. ran x ) ) ) = ( x e. _V |-> ( |s " ( ~P U. ran x X. ~P U. ran x ) ) )
28	26, 27	fvmptg 6280	. . 3 |- ( ( ( M |` A ) e. _V /\ ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) e. _V ) -> ( ( x e. _V |-> ( |s " ( ~P U. ran x X. ~P U. ran x ) ) ) ` ( M |` A ) ) = ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) )
29	7, 19, 28	syl2anc 693	. 2 |- ( A e. On -> ( ( x e. _V |-> ( |s " ( ~P U. ran x X. ~P U. ran x ) ) ) ` ( M |` A ) ) = ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) )
30	2, 29	eqtrd 2656	1 |- ( A e. On -> ( M ` A ) = ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   ran crn 5115   |` cres 5116   "cima 5117   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   Nocsur 31793   < <scsslt 31896   |scscut 31898   M cmade 31925

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-3or 1038  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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798  df-sslt 31897  df-scut 31899  df-made 31930

This theorem is referenced by:  madeval2  31936