Theorem om2uzsuci 12747

Description: The value of G (see om2uz0i 12746) at a successor. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Hypotheses

Ref	Expression
om2uz.1	|- C e. ZZ
om2uz.2	|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )

Assertion

Ref	Expression
om2uzsuci	|- ( A e. om -> ( G ` suc A ) = ( ( G ` A ) + 1 ) )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   G( x)

Proof of Theorem om2uzsuci

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suceq 5790	. . . 4 |- ( z = A -> suc z = suc A )
2	1	fveq2d 6195	. . 3 |- ( z = A -> ( G ` suc z ) = ( G ` suc A ) )
3		fveq2 6191	. . . 4 |- ( z = A -> ( G ` z ) = ( G ` A ) )
4	3	oveq1d 6665	. . 3 |- ( z = A -> ( ( G ` z ) + 1 ) = ( ( G ` A ) + 1 ) )
5	2, 4	eqeq12d 2637	. 2 |- ( z = A -> ( ( G ` suc z ) = ( ( G ` z ) + 1 ) <-> ( G ` suc A ) = ( ( G ` A ) + 1 ) ) )
6		ovex 6678	. . 3 |- ( ( G ` z ) + 1 ) e. _V
7		om2uz.2	. . . 4 |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )
8		oveq1 6657	. . . 4 |- ( y = x -> ( y + 1 ) = ( x + 1 ) )
9		oveq1 6657	. . . 4 |- ( y = ( G ` z ) -> ( y + 1 ) = ( ( G ` z ) + 1 ) )
10	7, 8, 9	frsucmpt2 7535	. . 3 |- ( ( z e. om /\ ( ( G ` z ) + 1 ) e. _V ) -> ( G ` suc z ) = ( ( G ` z ) + 1 ) )
11	6, 10	mpan2 707	. 2 |- ( z e. om -> ( G ` suc z ) = ( ( G ` z ) + 1 ) )
12	5, 11	vtoclga 3272	1 |- ( A e. om -> ( G ` suc A ) = ( ( G ` A ) + 1 ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   |` cres 5116   suc csuc 5725   ` cfv 5888  (class class class)co 6650   omcom 7065   reccrdg 7505   1c1 9937   + caddc 9939   ZZcz 11377

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-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-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-lim 5728  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-ov 6653  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  om2uzuzi  12748  om2uzlti  12749  om2uzrani  12751  om2uzrdg  12755  uzrdgsuci  12759  uzrdgxfr  12766  fzennn  12767  axdc4uzlem  12782  hashgadd  13166