Theorem frfnom 3951

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 3193	. 2 |- ((rec(F, A) |` om) Fn om <-> (Fun (rec(F, A) |` om) /\ dom (rec(F, A) |` om) = om))
2		rdgfnon 3939	. . . 4 |- rec(F, A) Fn On
3		fnfun 3585	. . . 4 |- (rec(F, A) Fn On -> Fun rec(F, A))
4	2, 3	ax-mp 7	. . 3 |- Fun rec(F, A)
5		funres 3551	. . 3 |- (Fun rec(F, A) -> Fun (rec(F, A) |` om))
6	4, 5	ax-mp 7	. 2 |- Fun (rec(F, A) |` om)
7		fndm 3587	. . . . 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 2214	. . 3 |- (om i^i dom rec(F, A)) = (om i^i On)
10		dmres 3380	. . 3 |- dom (rec(F, A) |` om) = (om i^i dom rec(F, A))
11		omsson 3136	. . . 4 |- om (_ On
12		dfss 2054	. . . 4 |- (om (_ On <-> om = (om i^i On))
13	11, 12	mpbi 189	. . 3 |- om = (om i^i On)
14	9, 10, 13	3eqtr4 1505	. 2 |- dom (rec(F, A) |` om) = om
15	1, 6, 14	mpbir2an 730	1 |- (rec(F, A) |` om) Fn om

Syntax hints:   = wceq 956   i^i cin 2046   (_ wss 2047  Oncon0 2948  omcom 3131  dom cdm 3170   |` cres 3172  Fun wfun 3176   Fn wfn 3177  reccrdg 3931

This theorem is referenced by:  unblem4 4543  inf0 4606  inf3lem6 4618  alephfplem4 4899  alephfp 4900  om2uzran 6300  om2uzf1o 6301

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932

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