Theorem om2uz0 4651

Description: The mapping G is a one-to-one mapping from om onto an upper partition of ZZ that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number C (normally 0 for the upper partition NN0 or 1 for the upper partition NN), 1 maps to C + 1, etc. This theorem shows the value of G at ordinal natural number zero. (Another version of this series of theorems was contributed by Raph Levien, 10-Apr-04.)

Hypotheses

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

Assertion

Ref	Expression
om2uz0	|- (G` (/)) = C

Distinct variable group(s):   x,y,C

Proof of Theorem om2uz0

Step	Hyp	Ref	Expression
1		om2uz.2	. . 3 |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)
2	1	fveq1i 2833	. 2 |- (G` (/)) = ((rec({<.x, y>. | y = (x + 1)}, C) |` om)` (/))
3		om2uz.1	. . 3 |- C e. ZZ
4		frzer 2990	. . 3 |- (C e. ZZ -> ((rec({<.x, y>. | y = (x + 1)}, C) |` om)` (/)) = C)
5	3, 4	ax-mp 6	. 2 |- ((rec({<.x, y>. | y = (x + 1)}, C) |` om)` (/)) = C
6	2, 5	eqtr 1119	1 |- (G` (/)) = C

Syntax hints:   = wceq 1091   e. wcel 1092  (/)c0 1707  {copab 2055  omcom 2372   |` cres 2412  ` cfv 2422  reccrdg 2969  (class class class)co 3001  1c1 4029   + caddc 4031  ZZcz 4095

This theorem is referenced by:  om2uzuz 4653  om2uzran 4655  uzrdgini 4658

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970

metamath.org