Theorem frfnom 5159

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 4009	. 2 |- ((rec(F, A) |` om) Fn om <-> (Fun (rec(F, A) |` om) /\ dom (rec(F, A) |` om) = om))
2		rdgfnon 5147	. . . 4 |- rec(F, A) Fn On
3		fnfun 4510	. . . 4 |- (rec(F, A) Fn On -> Fun rec(F, A))
4	2, 3	ax-mp 7	. . 3 |- Fun rec(F, A)
5		funres 4459	. . 3 |- (Fun rec(F, A) -> Fun (rec(F, A) |` om))
6	4, 5	ax-mp 7	. 2 |- Fun (rec(F, A) |` om)
7		fndm 4512	. . . . 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 2793	. . 3 |- (om i^i dom rec(F, A)) = (om i^i On)
10		dmres 4234	. . 3 |- dom (rec(F, A) |` om) = (om i^i dom rec(F, A))
11		omsson 3954	. . . 4 |- om C_ On
12		dfss 2606	. . . 4 |- (om C_ On <-> om = (om i^i On))
13	11, 12	mpbi 206	. . 3 |- om = (om i^i On)
14	9, 10, 13	3eqtr4i 1921	. 2 |- dom (rec(F, A) |` om) = om
15	1, 6, 14	mpbir2an 800	1 |- (rec(F, A) |` om) Fn om

Syntax hints:   = wceq 1298   i^i cin 2592   C_ wss 2593  Oncon0 3657  omcom 3949  dom cdm 3986   |` cres 3988  Fun wfun 3992   Fn wfn 3993  reccrdg 5139

This theorem is referenced by:  unblem4 5636  inf0 5712  inf3lem6 5724  alephfplem4 6047  alephfp 6048  om2uzrani 7711  om2uzf1oi 7712  frsucopabn 13911  eltrcl 13932  trclpred 13934  trclss 13935  trcltr 13936  trclex 13937

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140

Copyright terms: Public domain