Theorem frec2uz0d 10785

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 5676	. 2 |- ( G ` (/) ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , C ) ` (/) )
3		frec2uz.1	. . 3 |- ( ph -> C e. ZZ )
4		frec0g 6641	. . 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 2279	1 |- ( ph -> ( G ` (/) ) = C )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   (/)c0 3512   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058  freccfrec 6634   1c1 8144   + caddc 8146   ZZcz 9594

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-recs 6549  df-frec 6635

This theorem is referenced by:  frec2uzuzd  10788  frec2uzrand  10791  frec2uzrdg  10795  frecuzrdgg  10802  frecfzennn  10812  0tonninf  10826  1tonninf  10827  omgadd  11191  ennnfonelem1  13242  ennnfonelemhf1o  13248  012of  16893  2o01f  16894  isomninnlem  16940  iswomninnlem  16960  ismkvnnlem  16963