Theorem frec2uz0d 10621

Description: The mapping G is a one-to-one mapping from om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number C (normally 0 for the upper integers NN0 or 1 for the upper integers NN), 1 maps to C + 1, etc. This theorem shows the value of G at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.)

Hypotheses

Ref	Expression
frec2uz.1	|- ( ph -> C e. ZZ )
frec2uz.2	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )

Assertion

Ref	Expression
frec2uz0d	|- ( ph -> ( G ` (/) ) = C )

Distinct variable group:   x, C

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

Proof of Theorem frec2uz0d

Step	Hyp	Ref	Expression
1		frec2uz.2	. . 3 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
2	1	fveq1i 5628	. 2 |- ( G ` (/) ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , C ) ` (/) )
3		frec2uz.1	. . 3 |- ( ph -> C e. ZZ )
4		frec0g 6543	. . 3 |- ( C e. ZZ -> (frec ( ( x e. ZZ |-> ( x + 1 ) ) , C ) ` (/) ) = C )
5	3, 4	syl 14	. 2 |- ( ph -> (frec ( ( x e. ZZ |-> ( x + 1 ) ) , C ) ` (/) ) = C )
6	2, 5	eqtrid 2274	1 |- ( ph -> ( G ` (/) ) = C )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   (/)c0 3491   |-> cmpt 4145   ` cfv 5318  (class class class)co 6001  freccfrec 6536   1c1 8000   + caddc 8002   ZZcz 9446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-recs 6451  df-frec 6537

This theorem is referenced by:  frec2uzuzd  10624  frec2uzrand  10627  frec2uzrdg  10631  frecuzrdgg  10638  frecfzennn  10648  0tonninf  10662  1tonninf  10663  omgadd  11024  ennnfonelem1  12978  ennnfonelemhf1o  12984  012of  16357  2o01f  16358  isomninnlem  16398  iswomninnlem  16417  ismkvnnlem  16420