Theorem frfnom 3946

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 3189	. 2 |- ((rec(F, A) |` om) Fn om <-> (Fun (rec(F, A) |` om) /\ dom (rec(F, A) |` om) = om))
2		rdgfnon 3934	. . . 4 |- rec(F, A) Fn On
3		fnfun 3581	. . . 4 |- (rec(F, A) Fn On -> Fun rec(F, A))
4	2, 3	ax-mp 7	. . 3 |- Fun rec(F, A)
5		funres 3547	. . 3 |- (Fun rec(F, A) -> Fun (rec(F, A) |` om))
6	4, 5	ax-mp 7	. 2 |- Fun (rec(F, A) |` om)
7		fndm 3583	. . . . 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 2211	. . 3 |- (om i^i dom rec(F, A)) = (om i^i On)
10		dmres 3376	. . 3 |- dom (rec(F, A) |` om) = (om i^i dom rec(F, A))
11		omsson 3132	. . . 4 |- om (_ On
12		dfss 2051	. . . 4 |- (om (_ On <-> om = (om i^i On))
13	11, 12	mpbi 189	. . 3 |- om = (om i^i On)
14	9, 10, 13	3eqtr4 1503	. 2 |- dom (rec(F, A) |` om) = om
15	1, 6, 14	mpbir2an 729	1 |- (rec(F, A) |` om) Fn om

Syntax hints:   = wceq 955   i^i cin 2043   (_ wss 2044  Oncon0 2944  omcom 3127  dom cdm 3166   |` cres 3168  Fun wfun 3172   Fn wfn 3173  reccrdg 3926

This theorem is referenced by:  unblem4 4529  inf0 4589  inf3lem6 4601  alephfplem4 4882  alephfp 4883  om2uzran 6250  om2uzf1o 6251

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194  df-rdg 3927

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