Theorem dnnumch3lem 37616

Description: Value of the ordinal injection 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
dnnumch3lem	|- ( ( ph /\ w e. A ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) = |^| ( `' F " { w } ) )

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

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

Proof of Theorem dnnumch3lem

Step	Hyp	Ref	Expression
1		simpr 477	. 2 |- ( ( ph /\ w e. A ) -> w e. A )
2		cnvimass 5485	. . . 4 |- ( `' F " { w } ) C_ dom F
3		dnnumch.f	. . . . . 6 |- F = recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) )
4	3	tfr1 7493	. . . . 5 |- F Fn On
5		fndm 5990	. . . . 5 |- ( F Fn On -> dom F = On )
6	4, 5	ax-mp 5	. . . 4 |- dom F = On
7	2, 6	sseqtri 3637	. . 3 |- ( `' F " { w } ) C_ On
8		dnnumch.a	. . . . . 6 |- ( ph -> A e. V )
9		dnnumch.g	. . . . . 6 |- ( ph -> A. y e. ~P A ( y =/= (/) -> ( G ` y ) e. y ) )
10	3, 8, 9	dnnumch2 37615	. . . . 5 |- ( ph -> A C_ ran F )
11	10	sselda 3603	. . . 4 |- ( ( ph /\ w e. A ) -> w e. ran F )
12		inisegn0 5497	. . . 4 |- ( w e. ran F <-> ( `' F " { w } ) =/= (/) )
13	11, 12	sylib 208	. . 3 |- ( ( ph /\ w e. A ) -> ( `' F " { w } ) =/= (/) )
14		oninton 7000	. . 3 |- ( ( ( `' F " { w } ) C_ On /\ ( `' F " { w } ) =/= (/) ) -> |^| ( `' F " { w } ) e. On )
15	7, 13, 14	sylancr 695	. 2 |- ( ( ph /\ w e. A ) -> |^| ( `' F " { w } ) e. On )
16		sneq 4187	. . . . 5 |- ( x = w -> { x } = { w } )
17	16	imaeq2d 5466	. . . 4 |- ( x = w -> ( `' F " { x } ) = ( `' F " { w } ) )
18	17	inteqd 4480	. . 3 |- ( x = w -> |^| ( `' F " { x } ) = |^| ( `' F " { w } ) )
19		eqid 2622	. . 3 |- ( x e. A |-> |^| ( `' F " { x } ) ) = ( x e. A |-> |^| ( `' F " { x } ) )
20	18, 19	fvmptg 6280	. 2 |- ( ( w e. A /\ |^| ( `' F " { w } ) e. On ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) = |^| ( `' F " { w } ) )
21	1, 15, 20	syl2anc 693	1 |- ( ( ph /\ w e. A ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) = |^| ( `' F " { w } ) )

Syntax hints:   -> wi 4   /\ 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   ` 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:  dnnumch3  37617  dnwech  37618