Theorem dnnumch3 37617

Description: Define an injection from a set into the ordinals using 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 ) )

Assertion

Ref	Expression
dnnumch3	|- ( ph -> ( x e. A |-> |^| ( `' F " { x } ) ) : A -1-1-> On )

Distinct variable groups:   x, F, y   x, G, y, z   x, A, y, z   ph, x

Allowed substitution hints:   ph( y, z)   F( z)   V( x, y, z)

Proof of Theorem dnnumch3

Dummy variables v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvimass 5485	. . . . 5 |- ( `' F " { x } ) C_ dom F
2		dnnumch.f	. . . . . . 7 |- F = recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) )
3	2	tfr1 7493	. . . . . 6 |- F Fn On
4		fndm 5990	. . . . . 6 |- ( F Fn On -> dom F = On )
5	3, 4	ax-mp 5	. . . . 5 |- dom F = On
6	1, 5	sseqtri 3637	. . . 4 |- ( `' F " { x } ) C_ On
7		dnnumch.a	. . . . . . 7 |- ( ph -> A e. V )
8		dnnumch.g	. . . . . . 7 |- ( ph -> A. y e. ~P A ( y =/= (/) -> ( G ` y ) e. y ) )
9	2, 7, 8	dnnumch2 37615	. . . . . 6 |- ( ph -> A C_ ran F )
10	9	sselda 3603	. . . . 5 |- ( ( ph /\ x e. A ) -> x e. ran F )
11		inisegn0 5497	. . . . 5 |- ( x e. ran F <-> ( `' F " { x } ) =/= (/) )
12	10, 11	sylib 208	. . . 4 |- ( ( ph /\ x e. A ) -> ( `' F " { x } ) =/= (/) )
13		oninton 7000	. . . 4 |- ( ( ( `' F " { x } ) C_ On /\ ( `' F " { x } ) =/= (/) ) -> |^| ( `' F " { x } ) e. On )
14	6, 12, 13	sylancr 695	. . 3 |- ( ( ph /\ x e. A ) -> |^| ( `' F " { x } ) e. On )
15		eqid 2622	. . 3 |- ( x e. A |-> |^| ( `' F " { x } ) ) = ( x e. A |-> |^| ( `' F " { x } ) )
16	14, 15	fmptd 6385	. 2 |- ( ph -> ( x e. A |-> |^| ( `' F " { x } ) ) : A --> On )
17	2, 7, 8	dnnumch3lem 37616	. . . . . 6 |- ( ( ph /\ v e. A ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) = |^| ( `' F " { v } ) )
18	17	adantrr 753	. . . . 5 |- ( ( ph /\ ( v e. A /\ w e. A ) ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) = |^| ( `' F " { v } ) )
19	2, 7, 8	dnnumch3lem 37616	. . . . . 6 |- ( ( ph /\ w e. A ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) = |^| ( `' F " { w } ) )
20	19	adantrl 752	. . . . 5 |- ( ( ph /\ ( v e. A /\ w e. A ) ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) = |^| ( `' F " { w } ) )
21	18, 20	eqeq12d 2637	. . . 4 |- ( ( ph /\ ( v e. A /\ w e. A ) ) -> ( ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) = ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) <-> |^| ( `' F " { v } ) = |^| ( `' F " { w } ) ) )
22		fveq2 6191	. . . . . . 7 |- ( |^| ( `' F " { v } ) = |^| ( `' F " { w } ) -> ( F ` |^| ( `' F " { v } ) ) = ( F ` |^| ( `' F " { w } ) ) )
23	22	adantl 482	. . . . . 6 |- ( ( ( ph /\ ( v e. A /\ w e. A ) ) /\ |^| ( `' F " { v } ) = |^| ( `' F " { w } ) ) -> ( F ` |^| ( `' F " { v } ) ) = ( F ` |^| ( `' F " { w } ) ) )
24		cnvimass 5485	. . . . . . . . . . 11 |- ( `' F " { v } ) C_ dom F
25	24, 5	sseqtri 3637	. . . . . . . . . 10 |- ( `' F " { v } ) C_ On
26	9	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ v e. A ) -> v e. ran F )
27		inisegn0 5497	. . . . . . . . . . 11 |- ( v e. ran F <-> ( `' F " { v } ) =/= (/) )
28	26, 27	sylib 208	. . . . . . . . . 10 |- ( ( ph /\ v e. A ) -> ( `' F " { v } ) =/= (/) )
29		onint 6995	. . . . . . . . . 10 |- ( ( ( `' F " { v } ) C_ On /\ ( `' F " { v } ) =/= (/) ) -> |^| ( `' F " { v } ) e. ( `' F " { v } ) )
30	25, 28, 29	sylancr 695	. . . . . . . . 9 |- ( ( ph /\ v e. A ) -> |^| ( `' F " { v } ) e. ( `' F " { v } ) )
31		fniniseg 6338	. . . . . . . . . . 11 |- ( F Fn On -> ( |^| ( `' F " { v } ) e. ( `' F " { v } ) <-> ( |^| ( `' F " { v } ) e. On /\ ( F ` |^| ( `' F " { v } ) ) = v ) ) )
32	3, 31	ax-mp 5	. . . . . . . . . 10 |- ( |^| ( `' F " { v } ) e. ( `' F " { v } ) <-> ( |^| ( `' F " { v } ) e. On /\ ( F ` |^| ( `' F " { v } ) ) = v ) )
33	32	simprbi 480	. . . . . . . . 9 |- ( |^| ( `' F " { v } ) e. ( `' F " { v } ) -> ( F ` |^| ( `' F " { v } ) ) = v )
34	30, 33	syl 17	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> ( F ` |^| ( `' F " { v } ) ) = v )
35	34	adantrr 753	. . . . . . 7 |- ( ( ph /\ ( v e. A /\ w e. A ) ) -> ( F ` |^| ( `' F " { v } ) ) = v )
36	35	adantr 481	. . . . . 6 |- ( ( ( ph /\ ( v e. A /\ w e. A ) ) /\ |^| ( `' F " { v } ) = |^| ( `' F " { w } ) ) -> ( F ` |^| ( `' F " { v } ) ) = v )
37		cnvimass 5485	. . . . . . . . . . 11 |- ( `' F " { w } ) C_ dom F
38	37, 5	sseqtri 3637	. . . . . . . . . 10 |- ( `' F " { w } ) C_ On
39	9	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ w e. A ) -> w e. ran F )
40		inisegn0 5497	. . . . . . . . . . 11 |- ( w e. ran F <-> ( `' F " { w } ) =/= (/) )
41	39, 40	sylib 208	. . . . . . . . . 10 |- ( ( ph /\ w e. A ) -> ( `' F " { w } ) =/= (/) )
42		onint 6995	. . . . . . . . . 10 |- ( ( ( `' F " { w } ) C_ On /\ ( `' F " { w } ) =/= (/) ) -> |^| ( `' F " { w } ) e. ( `' F " { w } ) )
43	38, 41, 42	sylancr 695	. . . . . . . . 9 |- ( ( ph /\ w e. A ) -> |^| ( `' F " { w } ) e. ( `' F " { w } ) )
44		fniniseg 6338	. . . . . . . . . . 11 |- ( F Fn On -> ( |^| ( `' F " { w } ) e. ( `' F " { w } ) <-> ( |^| ( `' F " { w } ) e. On /\ ( F ` |^| ( `' F " { w } ) ) = w ) ) )
45	3, 44	ax-mp 5	. . . . . . . . . 10 |- ( |^| ( `' F " { w } ) e. ( `' F " { w } ) <-> ( |^| ( `' F " { w } ) e. On /\ ( F ` |^| ( `' F " { w } ) ) = w ) )
46	45	simprbi 480	. . . . . . . . 9 |- ( |^| ( `' F " { w } ) e. ( `' F " { w } ) -> ( F ` |^| ( `' F " { w } ) ) = w )
47	43, 46	syl 17	. . . . . . . 8 |- ( ( ph /\ w e. A ) -> ( F ` |^| ( `' F " { w } ) ) = w )
48	47	adantrl 752	. . . . . . 7 |- ( ( ph /\ ( v e. A /\ w e. A ) ) -> ( F ` |^| ( `' F " { w } ) ) = w )
49	48	adantr 481	. . . . . 6 |- ( ( ( ph /\ ( v e. A /\ w e. A ) ) /\ |^| ( `' F " { v } ) = |^| ( `' F " { w } ) ) -> ( F ` |^| ( `' F " { w } ) ) = w )
50	23, 36, 49	3eqtr3d 2664	. . . . 5 |- ( ( ( ph /\ ( v e. A /\ w e. A ) ) /\ |^| ( `' F " { v } ) = |^| ( `' F " { w } ) ) -> v = w )
51	50	ex 450	. . . 4 |- ( ( ph /\ ( v e. A /\ w e. A ) ) -> ( |^| ( `' F " { v } ) = |^| ( `' F " { w } ) -> v = w ) )
52	21, 51	sylbid 230	. . 3 |- ( ( ph /\ ( v e. A /\ w e. A ) ) -> ( ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) = ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) -> v = w ) )
53	52	ralrimivva 2971	. 2 |- ( ph -> A. v e. A A. w e. A ( ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) = ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) -> v = w ) )
54		dff13 6512	. 2 |- ( ( x e. A |-> |^| ( `' F " { x } ) ) : A -1-1-> On <-> ( ( x e. A |-> |^| ( `' F " { x } ) ) : A --> On /\ A. v e. A A. w e. A ( ( ( x e. A |-> |^| ( `' F " { x } ) ) ` v ) = ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) -> v = w ) ) )
55	16, 53, 54	sylanbrc 698	1 |- ( ph -> ( x e. A |-> |^| ( `' F " { x } ) ) : A -1-1-> On )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   |^|cint 4475   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Oncon0 5723   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   ` 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-wrecs 7407  df-recs 7468

This theorem is referenced by:  dnwech  37618