Theorem dnnumch1 37614

Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 8853. (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
dnnumch1	|- ( ph -> E. x e. On ( F |` x ) : x -1-1-onto-> A )

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 dnnumch1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dnnumch.a	. 2 |- ( ph -> A e. V )
2		recsval 7500	. . . . . . 7 |- ( x e. On -> (recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ) ` x ) = ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ` (recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ) |` x ) ) )
3		dnnumch.f	. . . . . . . 8 |- F = recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) )
4	3	fveq1i 6192	. . . . . . 7 |- ( F ` x ) = (recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ) ` x )
5	3	tfr1 7493	. . . . . . . . . . 11 |- F Fn On
6		fnfun 5988	. . . . . . . . . . 11 |- ( F Fn On -> Fun F )
7	5, 6	ax-mp 5	. . . . . . . . . 10 |- Fun F
8		vex 3203	. . . . . . . . . 10 |- x e. _V
9		resfunexg 6479	. . . . . . . . . 10 |- ( ( Fun F /\ x e. _V ) -> ( F |` x ) e. _V )
10	7, 8, 9	mp2an 708	. . . . . . . . 9 |- ( F |` x ) e. _V
11		rneq 5351	. . . . . . . . . . . . 13 |- ( w = ( F |` x ) -> ran w = ran ( F |` x ) )
12		df-ima 5127	. . . . . . . . . . . . 13 |- ( F " x ) = ran ( F |` x )
13	11, 12	syl6eqr 2674	. . . . . . . . . . . 12 |- ( w = ( F |` x ) -> ran w = ( F " x ) )
14	13	difeq2d 3728	. . . . . . . . . . 11 |- ( w = ( F |` x ) -> ( A \ ran w ) = ( A \ ( F " x ) ) )
15	14	fveq2d 6195	. . . . . . . . . 10 |- ( w = ( F |` x ) -> ( G ` ( A \ ran w ) ) = ( G ` ( A \ ( F " x ) ) ) )
16		rneq 5351	. . . . . . . . . . . . 13 |- ( z = w -> ran z = ran w )
17	16	difeq2d 3728	. . . . . . . . . . . 12 |- ( z = w -> ( A \ ran z ) = ( A \ ran w ) )
18	17	fveq2d 6195	. . . . . . . . . . 11 |- ( z = w -> ( G ` ( A \ ran z ) ) = ( G ` ( A \ ran w ) ) )
19	18	cbvmptv 4750	. . . . . . . . . 10 |- ( z e. _V |-> ( G ` ( A \ ran z ) ) ) = ( w e. _V |-> ( G ` ( A \ ran w ) ) )
20		fvex 6201	. . . . . . . . . 10 |- ( G ` ( A \ ( F " x ) ) ) e. _V
21	15, 19, 20	fvmpt 6282	. . . . . . . . 9 |- ( ( F |` x ) e. _V -> ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ` ( F |` x ) ) = ( G ` ( A \ ( F " x ) ) ) )
22	10, 21	ax-mp 5	. . . . . . . 8 |- ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ` ( F |` x ) ) = ( G ` ( A \ ( F " x ) ) )
23	3	reseq1i 5392	. . . . . . . . 9 |- ( F |` x ) = (recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ) |` x )
24	23	fveq2i 6194	. . . . . . . 8 |- ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ` ( F |` x ) ) = ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ` (recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ) |` x ) )
25	22, 24	eqtr3i 2646	. . . . . . 7 |- ( G ` ( A \ ( F " x ) ) ) = ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ` (recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ) |` x ) )
26	2, 4, 25	3eqtr4g 2681	. . . . . 6 |- ( x e. On -> ( F ` x ) = ( G ` ( A \ ( F " x ) ) ) )
27	26	ad2antlr 763	. . . . 5 |- ( ( ( ph /\ x e. On ) /\ ( A \ ( F " x ) ) =/= (/) ) -> ( F ` x ) = ( G ` ( A \ ( F " x ) ) ) )
28		difss 3737	. . . . . . . . 9 |- ( A \ ( F " x ) ) C_ A
29		elpw2g 4827	. . . . . . . . . 10 |- ( A e. V -> ( ( A \ ( F " x ) ) e. ~P A <-> ( A \ ( F " x ) ) C_ A ) )
30	1, 29	syl 17	. . . . . . . . 9 |- ( ph -> ( ( A \ ( F " x ) ) e. ~P A <-> ( A \ ( F " x ) ) C_ A ) )
31	28, 30	mpbiri 248	. . . . . . . 8 |- ( ph -> ( A \ ( F " x ) ) e. ~P A )
32		dnnumch.g	. . . . . . . 8 |- ( ph -> A. y e. ~P A ( y =/= (/) -> ( G ` y ) e. y ) )
33		neeq1 2856	. . . . . . . . . 10 |- ( y = ( A \ ( F " x ) ) -> ( y =/= (/) <-> ( A \ ( F " x ) ) =/= (/) ) )
34		fveq2 6191	. . . . . . . . . . 11 |- ( y = ( A \ ( F " x ) ) -> ( G ` y ) = ( G ` ( A \ ( F " x ) ) ) )
35		id 22	. . . . . . . . . . 11 |- ( y = ( A \ ( F " x ) ) -> y = ( A \ ( F " x ) ) )
36	34, 35	eleq12d 2695	. . . . . . . . . 10 |- ( y = ( A \ ( F " x ) ) -> ( ( G ` y ) e. y <-> ( G ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) ) )
37	33, 36	imbi12d 334	. . . . . . . . 9 |- ( y = ( A \ ( F " x ) ) -> ( ( y =/= (/) -> ( G ` y ) e. y ) <-> ( ( A \ ( F " x ) ) =/= (/) -> ( G ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) ) ) )
38	37	rspcva 3307	. . . . . . . 8 |- ( ( ( A \ ( F " x ) ) e. ~P A /\ A. y e. ~P A ( y =/= (/) -> ( G ` y ) e. y ) ) -> ( ( A \ ( F " x ) ) =/= (/) -> ( G ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) ) )
39	31, 32, 38	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( A \ ( F " x ) ) =/= (/) -> ( G ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) ) )
40	39	adantr 481	. . . . . 6 |- ( ( ph /\ x e. On ) -> ( ( A \ ( F " x ) ) =/= (/) -> ( G ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) ) )
41	40	imp 445	. . . . 5 |- ( ( ( ph /\ x e. On ) /\ ( A \ ( F " x ) ) =/= (/) ) -> ( G ` ( A \ ( F " x ) ) ) e. ( A \ ( F " x ) ) )
42	27, 41	eqeltrd 2701	. . . 4 |- ( ( ( ph /\ x e. On ) /\ ( A \ ( F " x ) ) =/= (/) ) -> ( F ` x ) e. ( A \ ( F " x ) ) )
43	42	ex 450	. . 3 |- ( ( ph /\ x e. On ) -> ( ( A \ ( F " x ) ) =/= (/) -> ( F ` x ) e. ( A \ ( F " x ) ) ) )
44	43	ralrimiva 2966	. 2 |- ( ph -> A. x e. On ( ( A \ ( F " x ) ) =/= (/) -> ( F ` x ) e. ( A \ ( F " x ) ) ) )
45	5	tz7.49c 7541	. 2 |- ( ( A e. V /\ A. x e. On ( ( A \ ( F " x ) ) =/= (/) -> ( F ` x ) e. ( A \ ( F " x ) ) ) ) -> E. x e. On ( F |` x ) : x -1-1-onto-> A )
46	1, 44, 45	syl2anc 693	1 |- ( ph -> E. x e. On ( F |` x ) : x -1-1-onto-> A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |-> cmpt 4729   ran crn 5115   |` cres 5116   "cima 5117   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   -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-wrecs 7407  df-recs 7468

This theorem is referenced by:  dnnumch2  37615