Theorem frec2uz0d 10508

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 5562	. 2 |- ( G ` (/) ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , C ) ` (/) )
3		frec2uz.1	. . 3 |- ( ph -> C e. ZZ )
4		frec0g 6464	. . 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 2241	1 |- ( ph -> ( G ` (/) ) = C )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   (/)c0 3451   |-> cmpt 4095   ` cfv 5259  (class class class)co 5925  freccfrec 6457   1c1 7897   + caddc 7899   ZZcz 9343

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-recs 6372  df-frec 6458

This theorem is referenced by:  frec2uzuzd  10511  frec2uzrand  10514  frec2uzrdg  10518  frecuzrdgg  10525  frecfzennn  10535  0tonninf  10549  1tonninf  10550  omgadd  10911  ennnfonelem1  12649  ennnfonelemhf1o  12655  012of  15724  2o01f  15725  isomninnlem  15761  iswomninnlem  15780  ismkvnnlem  15783