Theorem frfnom 3965

Description: The function generated by finite recursive definition generation is a function on omega.

Assertion

Ref	Expression
frfnom	|- (rec(F, A) |` om) Fn om

Proof of Theorem frfnom

Step	Hyp	Ref	Expression
1		df-fn 3207	. 2 |- ((rec(F, A) |` om) Fn om <-> (Fun (rec(F, A) |` om) /\ dom (rec(F, A) |` om) = om))
2		rdgfnon 3953	. . . 4 |- rec(F, A) Fn On
3		fnfun 3599	. . . 4 |- (rec(F, A) Fn On -> Fun rec(F, A))
4	2, 3	ax-mp 7	. . 3 |- Fun rec(F, A)
5		funres 3565	. . 3 |- (Fun rec(F, A) -> Fun (rec(F, A) |` om))
6	4, 5	ax-mp 7	. 2 |- Fun (rec(F, A) |` om)
7		fndm 3601	. . . . 5 |- (rec(F, A) Fn On -> dom rec(F, A) = On)
8	2, 7	ax-mp 7	. . . 4 |- dom rec(F, A) = On
9	8	ineq2i 2223	. . 3 |- (om i^i dom rec(F, A)) = (om i^i On)
10		dmres 3394	. . 3 |- dom (rec(F, A) |` om) = (om i^i dom rec(F, A))
11		omsson 3150	. . . 4 |- om (_ On
12		dfss 2063	. . . 4 |- (om (_ On <-> om = (om i^i On))
13	11, 12	mpbi 189	. . 3 |- om = (om i^i On)
14	9, 10, 13	3eqtr4 1512	. 2 |- dom (rec(F, A) |` om) = om
15	1, 6, 14	mpbir2an 734	1 |- (rec(F, A) |` om) Fn om

Syntax hints:   = wceq 960   i^i cin 2055   (_ wss 2056  Oncon0 2962  omcom 3145  dom cdm 3184   |` cres 3186  Fun wfun 3190   Fn wfn 3191  reccrdg 3945

This theorem is referenced by:  unblem4 4556  inf0 4618  inf3lem6 4630  alephfplem4 4912  alephfp 4913  om2uzran 6482  om2uzf1o 6483

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-rep 2706  ax-sep 2716  ax-nul 2723  ax-pow 2756  ax-pr 2793  ax-un 2880

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 780  df-3an 781  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-ral 1656  df-rex 1657  df-rab 1659  df-v 1819  df-sbc 1949  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-tp 2425  df-op 2426  df-uni 2516  df-iun 2580  df-br 2633  df-opab 2680  df-tr 2694  df-eprel 2846  df-id 2849  df-po 2854  df-so 2864  df-fr 2931  df-we 2948  df-ord 2965  df-on 2966  df-suc 2968  df-om 3146  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fn 3207  df-fv 3212  df-rdg 3946

Copyright terms: Public domain