Theorem madeval2 31936

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

Assertion

Ref	Expression
madeval2	|- ( A e. On -> ( M ` A ) = { x e. No | E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) } )

Distinct variable group:   x, A, a, b

Proof of Theorem madeval2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		madeval 31935	. 2 |- ( A e. On -> ( M ` A ) = ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) )
2		scutcut 31912	. . . . . . . . 9 |- ( a < <s b -> ( ( a |s b ) e. No /\ a < <s { ( a |s b ) } /\ { ( a |s b ) } < <s b ) )
3	2	simp1d 1073	. . . . . . . 8 |- ( a < <s b -> ( a |s b ) e. No )
4		eleq1 2689	. . . . . . . . 9 |- ( ( a |s b ) = x -> ( ( a |s b ) e. No <-> x e. No ) )
5	4	biimpd 219	. . . . . . . 8 |- ( ( a |s b ) = x -> ( ( a |s b ) e. No -> x e. No ) )
6	3, 5	mpan9 486	. . . . . . 7 |- ( ( a < <s b /\ ( a |s b ) = x ) -> x e. No )
7	6	rexlimivw 3029	. . . . . 6 |- ( E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) -> x e. No )
8	7	rexlimivw 3029	. . . . 5 |- ( E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) -> x e. No )
9	8	pm4.71ri 665	. . . 4 |- ( E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) <-> ( x e. No /\ E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) ) )
10	9	abbii 2739	. . 3 |- { x | E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) } = { x | ( x e. No /\ E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) ) }
11		eleq1 2689	. . . . . . 7 |- ( y = <. a , b >. -> ( y e. < <s <-> <. a , b >. e. < <s ) )
12		breq1 4656	. . . . . . 7 |- ( y = <. a , b >. -> ( y |s x <-> <. a , b >. |s x ) )
13	11, 12	anbi12d 747	. . . . . 6 |- ( y = <. a , b >. -> ( ( y e. < <s /\ y |s x ) <-> ( <. a , b >. e. < <s /\ <. a , b >. |s x ) ) )
14	13	rexxp 5264	. . . . 5 |- ( E. y e. ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ( y e. < <s /\ y |s x ) <-> E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( <. a , b >. e. < <s /\ <. a , b >. |s x ) )
15		imaindm 31682	. . . . . . . 8 |- ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) = ( |s " ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i dom |s ) )
16		dmscut 31918	. . . . . . . . . 10 |- dom |s = < <s
17	16	ineq2i 3811	. . . . . . . . 9 |- ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i dom |s ) = ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i < <s )
18	17	imaeq2i 5464	. . . . . . . 8 |- ( |s " ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i dom |s ) ) = ( |s " ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i < <s ) )
19	15, 18	eqtri 2644	. . . . . . 7 |- ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) = ( |s " ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i < <s ) )
20	19	eleq2i 2693	. . . . . 6 |- ( x e. ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) <-> x e. ( |s " ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i < <s ) ) )
21		vex 3203	. . . . . . 7 |- x e. _V
22	21	elima 5471	. . . . . 6 |- ( x e. ( |s " ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i < <s ) ) <-> E. y e. ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i < <s ) y |s x )
23		elin 3796	. . . . . . . . 9 |- ( y e. ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i < <s ) <-> ( y e. ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) /\ y e. < <s ) )
24	23	anbi1i 731	. . . . . . . 8 |- ( ( y e. ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i < <s ) /\ y |s x ) <-> ( ( y e. ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) /\ y e. < <s ) /\ y |s x ) )
25		anass 681	. . . . . . . 8 |- ( ( ( y e. ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) /\ y e. < <s ) /\ y |s x ) <-> ( y e. ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) /\ ( y e. < <s /\ y |s x ) ) )
26	24, 25	bitri 264	. . . . . . 7 |- ( ( y e. ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i < <s ) /\ y |s x ) <-> ( y e. ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) /\ ( y e. < <s /\ y |s x ) ) )
27	26	rexbii2 3039	. . . . . 6 |- ( E. y e. ( ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) i^i < <s ) y |s x <-> E. y e. ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ( y e. < <s /\ y |s x ) )
28	20, 22, 27	3bitri 286	. . . . 5 |- ( x e. ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) <-> E. y e. ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ( y e. < <s /\ y |s x ) )
29		df-br 4654	. . . . . . . 8 |- ( a < <s b <-> <. a , b >. e. < <s )
30	29	anbi1i 731	. . . . . . 7 |- ( ( a < <s b /\ ( a |s b ) = x ) <-> ( <. a , b >. e. < <s /\ ( a |s b ) = x ) )
31		df-ov 6653	. . . . . . . . . 10 |- ( a |s b ) = ( |s ` <. a , b >. )
32	31	eqeq1i 2627	. . . . . . . . 9 |- ( ( a |s b ) = x <-> ( |s ` <. a , b >. ) = x )
33		scutf 31919	. . . . . . . . . . 11 |- |s : < <s --> No
34		ffn 6045	. . . . . . . . . . 11 |- ( |s : < <s --> No -> |s Fn < <s )
35	33, 34	ax-mp 5	. . . . . . . . . 10 |- |s Fn < <s
36		fnbrfvb 6236	. . . . . . . . . 10 |- ( ( |s Fn < <s /\ <. a , b >. e. < <s ) -> ( ( |s ` <. a , b >. ) = x <-> <. a , b >. |s x ) )
37	35, 36	mpan 706	. . . . . . . . 9 |- ( <. a , b >. e. < <s -> ( ( |s ` <. a , b >. ) = x <-> <. a , b >. |s x ) )
38	32, 37	syl5bb 272	. . . . . . . 8 |- ( <. a , b >. e. < <s -> ( ( a |s b ) = x <-> <. a , b >. |s x ) )
39	38	pm5.32i 669	. . . . . . 7 |- ( ( <. a , b >. e. < <s /\ ( a |s b ) = x ) <-> ( <. a , b >. e. < <s /\ <. a , b >. |s x ) )
40	30, 39	bitri 264	. . . . . 6 |- ( ( a < <s b /\ ( a |s b ) = x ) <-> ( <. a , b >. e. < <s /\ <. a , b >. |s x ) )
41	40	2rexbii 3042	. . . . 5 |- ( E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) <-> E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( <. a , b >. e. < <s /\ <. a , b >. |s x ) )
42	14, 28, 41	3bitr4i 292	. . . 4 |- ( x e. ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) <-> E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) )
43	42	abbi2i 2738	. . 3 |- ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) = { x | E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) }
44		df-rab 2921	. . 3 |- { x e. No | E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) } = { x | ( x e. No /\ E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) ) }
45	10, 43, 44	3eqtr4i 2654	. 2 |- ( |s " ( ~P U. ( M " A ) X. ~P U. ( M " A ) ) ) = { x e. No | E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) }
46	1, 45	syl6eq 2672	1 |- ( A e. On -> ( M ` A ) = { x e. No | E. a e. ~P U. ( M " A ) E. b e. ~P U. ( M " A ) ( a < <s b /\ ( a |s b ) = x ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   i^i cin 3573   ~Pcpw 4158   {csn 4177   <.cop 4183   U.cuni 4436   class class class wbr 4653   X. cxp 5112   dom cdm 5114   "cima 5117   Oncon0 5723   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   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: (None)