Theorem frfnom 2989

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

Assertion

Ref	Expression
frfnom	⊢ (rec(F, A) ↾ ω) Fn ω

Proof of Theorem frfnom

Step	Hyp	Ref	Expression
1		rdgfnon 2977	. . . . 5 ⊢ rec(F, A) Fn On
2		fnfun 2721	. . . . 5 ⊢ (rec(F, A) Fn On → Fun rec(F, A))
3	1, 2	ax-mp 6	. . . 4 ⊢ Fun rec(F, A)
4		funres 2697	. . . 4 ⊢ (Fun rec(F, A) → Fun (rec(F, A) ↾ ω))
5	3, 4	ax-mp 6	. . 3 ⊢ Fun (rec(F, A) ↾ ω)
6		fndm 2723	. . . . . 6 ⊢ (rec(F, A) Fn On → dom rec(F, A) = On)
7	1, 6	ax-mp 6	. . . . 5 ⊢ dom rec(F, A) = On
8	7	ineq2i 1642	. . . 4 ⊢ (ω ∩ dom rec(F, A)) = (ω ∩ On)
9		dmres 2584	. . . 4 ⊢ dom (rec(F, A) ↾ ω) = (ω ∩ dom rec(F, A))
10		omsson 2377	. . . . 5 ⊢ ω ⊆ On
11		dfss 1493	. . . . 5 ⊢ (ω ⊆ On ↔ ω = (ω ∩ On))
12	10, 11	mpbi 164	. . . 4 ⊢ ω = (ω ∩ On)
13	8, 9, 12	3eqtr4 1126	. . 3 ⊢ dom (rec(F, A) ↾ ω) = ω
14	5, 13	pm3.2i 234	. 2 ⊢ (Fun (rec(F, A) ↾ ω) ∧ dom (rec(F, A) ↾ ω) = ω)
15		df-fn 2433	. 2 ⊢ ((rec(F, A) ↾ ω) Fn ω ↔ (Fun (rec(F, A) ↾ ω) ∧ dom (rec(F, A) ↾ ω) = ω))
16	14, 15	mpbir 165	1 ⊢ (rec(F, A) ↾ ω) Fn ω

Syntax hints:   ∧ wa 196   = wceq 1091   ∩ cin 1486   ⊆ wss 1487  Oncon0 2199  ωcom 2372  dom cdm 2410   ↾ cres 2412  Fun wfun 2416   Fn wfn 2417  reccrdg 2969

This theorem is referenced by:  unblem4 3434  inf0 3457  inf3lem6 3469  om2uzran 4655  om2uzf1o 4656

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-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

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