Theorem dnwech 37618

Description: Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypotheses

Ref	Expression
dnnumch.f	|- F = recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) )
dnnumch.a	|- ( ph -> A e. V )
dnnumch.g	|- ( ph -> A. y e. ~P A ( y =/= (/) -> ( G ` y ) e. y ) )
dnwech.h	|- H = { <. v , w >. | |^| ( `' F " { v } ) e. |^| ( `' F " { w } ) }

Assertion

Ref	Expression
dnwech	|- ( ph -> H We A )

Distinct variable groups:   v, F, w, y   v, G, w, y, z   v, A, w, y, z   ph, v, w

Allowed substitution hints:   ph( y, z)   F( z)   H( y, z, w, v)   V( y, z, w, v)

Proof of Theorem dnwech

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dnnumch.f	. . . . 5 |- F = recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) )
2		dnnumch.a	. . . . 5 |- ( ph -> A e. V )
3		dnnumch.g	. . . . 5 |- ( ph -> A. y e. ~P A ( y =/= (/) -> ( G ` y ) e. y ) )
4	1, 2, 3	dnnumch3 37617	. . . 4 |- ( ph -> ( x e. A |-> |^| ( `' F " { x } ) ) : A -1-1-> On )
5		f1f1orn 6148	. . . 4 |- ( ( x e. A |-> |^| ( `' F " { x } ) ) : A -1-1-> On -> ( x e. A |-> |^| ( `' F " { x } ) ) : A -1-1-onto-> ran ( x e. A |-> |^| ( `' F " { x } ) ) )
6	4, 5	syl 17	. . 3 |- ( ph -> ( x e. A |-> |^| ( `' F " { x } ) ) : A -1-1-onto-> ran ( x e. A |-> |^| ( `' F " { x } ) ) )
7		f1f 6101	. . . . 5 |- ( ( x e. A |-> |^| ( `' F " { x } ) ) : A -1-1-> On -> ( x e. A |-> |^| ( `' F " { x } ) ) : A --> On )
8		frn 6053	. . . . 5 |- ( ( x e. A |-> |^| ( `' F " { x } ) ) : A --> On -> ran ( x e. A |-> |^| ( `' F " { x } ) ) C_ On )
9	4, 7, 8	3syl 18	. . . 4 |- ( ph -> ran ( x e. A |-> |^| ( `' F " { x } ) ) C_ On )
10		epweon 6983	. . . 4 |- _E We On
11		wess 5101	. . . 4 |- ( ran ( x e. A |-> |^| ( `' F " { x } ) ) C_ On -> ( _E We On -> _E We ran ( x e. A |-> |^| ( `' F " { x } ) ) ) )
12	9, 10, 11	mpisyl 21	. . 3 |- ( ph -> _E We ran ( x e. A |-> |^| ( `' F " { x } ) ) )
13		eqid 2622	. . . 4 |- { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } = { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) }
14	13	f1owe 6603	. . 3 |- ( ( x e. A |-> |^| ( `' F " { x } ) ) : A -1-1-onto-> ran ( x e. A |-> |^| ( `' F " { x } ) ) -> ( _E We ran ( x e. A |-> |^| ( `' F " { x } ) ) -> { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } We A ) )
15	6, 12, 14	sylc 65	. 2 |- ( ph -> { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } We A )
16		fvex 6201	. . . . . . . . 9 |- ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) e. _V
17	16	epelc 5031	. . . . . . . 8 |- ( ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) <-> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) e. ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) )
18	1, 2, 3	dnnumch3lem 37616	. . . . . . . . . 10 |- ( ( ph /\ v e. A ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) = |^| ( `' F " { v } ) )
19	18	adantrr 753	. . . . . . . . 9 |- ( ( ph /\ ( v e. A /\ w e. A ) ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) = |^| ( `' F " { v } ) )
20	1, 2, 3	dnnumch3lem 37616	. . . . . . . . . 10 |- ( ( ph /\ w e. A ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) = |^| ( `' F " { w } ) )
21	20	adantrl 752	. . . . . . . . 9 |- ( ( ph /\ ( v e. A /\ w e. A ) ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) = |^| ( `' F " { w } ) )
22	19, 21	eleq12d 2695	. . . . . . . 8 |- ( ( ph /\ ( v e. A /\ w e. A ) ) -> ( ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) e. ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) <-> |^| ( `' F " { v } ) e. |^| ( `' F " { w } ) ) )
23	17, 22	syl5rbb 273	. . . . . . 7 |- ( ( ph /\ ( v e. A /\ w e. A ) ) -> ( |^| ( `' F " { v } ) e. |^| ( `' F " { w } ) <-> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) ) )
24	23	pm5.32da 673	. . . . . 6 |- ( ph -> ( ( ( v e. A /\ w e. A ) /\ |^| ( `' F " { v } ) e. |^| ( `' F " { w } ) ) <-> ( ( v e. A /\ w e. A ) /\ ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) ) ) )
25	24	opabbidv 4716	. . . . 5 |- ( ph -> { <. v , w >. | ( ( v e. A /\ w e. A ) /\ |^| ( `' F " { v } ) e. |^| ( `' F " { w } ) ) } = { <. v , w >. | ( ( v e. A /\ w e. A ) /\ ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) ) } )
26		incom 3805	. . . . . 6 |- ( H i^i ( A X. A ) ) = ( ( A X. A ) i^i H )
27		df-xp 5120	. . . . . . 7 |- ( A X. A ) = { <. v , w >. | ( v e. A /\ w e. A ) }
28		dnwech.h	. . . . . . 7 |- H = { <. v , w >. | |^| ( `' F " { v } ) e. |^| ( `' F " { w } ) }
29	27, 28	ineq12i 3812	. . . . . 6 |- ( ( A X. A ) i^i H ) = ( { <. v , w >. | ( v e. A /\ w e. A ) } i^i { <. v , w >. | |^| ( `' F " { v } ) e. |^| ( `' F " { w } ) } )
30		inopab 5252	. . . . . 6 |- ( { <. v , w >. | ( v e. A /\ w e. A ) } i^i { <. v , w >. | |^| ( `' F " { v } ) e. |^| ( `' F " { w } ) } ) = { <. v , w >. | ( ( v e. A /\ w e. A ) /\ |^| ( `' F " { v } ) e. |^| ( `' F " { w } ) ) }
31	26, 29, 30	3eqtri 2648	. . . . 5 |- ( H i^i ( A X. A ) ) = { <. v , w >. | ( ( v e. A /\ w e. A ) /\ |^| ( `' F " { v } ) e. |^| ( `' F " { w } ) ) }
32		incom 3805	. . . . . 6 |- ( { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } i^i ( A X. A ) ) = ( ( A X. A ) i^i { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } )
33	27	ineq1i 3810	. . . . . 6 |- ( ( A X. A ) i^i { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } ) = ( { <. v , w >. | ( v e. A /\ w e. A ) } i^i { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } )
34		inopab 5252	. . . . . 6 |- ( { <. v , w >. | ( v e. A /\ w e. A ) } i^i { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } ) = { <. v , w >. | ( ( v e. A /\ w e. A ) /\ ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) ) }
35	32, 33, 34	3eqtri 2648	. . . . 5 |- ( { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } i^i ( A X. A ) ) = { <. v , w >. | ( ( v e. A /\ w e. A ) /\ ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) ) }
36	25, 31, 35	3eqtr4g 2681	. . . 4 |- ( ph -> ( H i^i ( A X. A ) ) = ( { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } i^i ( A X. A ) ) )
37		weeq1 5102	. . . 4 |- ( ( H i^i ( A X. A ) ) = ( { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } i^i ( A X. A ) ) -> ( ( H i^i ( A X. A ) ) We A <-> ( { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } i^i ( A X. A ) ) We A ) )
38	36, 37	syl 17	. . 3 |- ( ph -> ( ( H i^i ( A X. A ) ) We A <-> ( { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } i^i ( A X. A ) ) We A ) )
39		weinxp 5186	. . 3 |- ( H We A <-> ( H i^i ( A X. A ) ) We A )
40		weinxp 5186	. . 3 |- ( { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } We A <-> ( { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } i^i ( A X. A ) ) We A )
41	38, 39, 40	3bitr4g 303	. 2 |- ( ph -> ( H We A <-> { <. v , w >. | ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) _E ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) } We A ) )
42	15, 41	mpbird 247	1 |- ( ph -> H We A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   |^|cint 4475   class class class wbr 4653   {copab 4712   |-> cmpt 4729   _E cep 5028   We wwe 5072   X. cxp 5112   `'ccnv 5113   ran crn 5115   "cima 5117   Oncon0 5723   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  recscrecs 7467

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-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-isom 5897  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  aomclem3  37626