Theorem frec2uz0d 9185

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 5179	. 2 |- ( G ` (/) ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , C ) ` (/) )
3		frec2uz.1	. . 3 |- ( ph -> C e. ZZ )
4		frec0g 5983	. . 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	syl5eq 2084	1 |- ( ph -> ( G ` (/) ) = C )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   (/)c0 3224   |-> cmpt 3818   ` cfv 4902  (class class class)co 5512  freccfrec 5977   1c1 6890   + caddc 6892   ZZcz 8245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-recs 5920  df-frec 5978

This theorem is referenced by:  frec2uzzd  9186  frec2uzuzd  9188  frec2uzrand  9191  frec2uzrdg  9195  frecfzennn  9203