Theorem rdglimt 3948

Description: The value of the recursive definition generator at a limit ordinal.

Assertion

Ref	Expression
rdglimt	|- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U.(rec(F, A)"B))

Proof of Theorem rdglimt

Step	Hyp	Ref	Expression
1		limelon 3032	. 2 |- ((B e. C /\ Lim B) -> B e. On)
2		limeq 2960	. . . . 5 |- (B = if(B e. On, B, (/)) -> (Lim B <-> Lim if(B e. On, B, (/))))
3		fveq2 3724	. . . . . 6 |- (B = if(B e. On, B, (/)) -> (rec(F, A)` B) = (rec(F, A)` if(B e. On, B, (/))))
4		imaeq2 3402	. . . . . . 7 |- (B = if(B e. On, B, (/)) -> (rec(F, A)"B) = (rec(F, A)"if(B e. On, B, (/))))
5	4	unieqd 2512	. . . . . 6 |- (B = if(B e. On, B, (/)) -> U.(rec(F, A)"B) = U.(rec(F, A)"if(B e. On, B, (/))))
6	3, 5	eqeq12d 1489	. . . . 5 |- (B = if(B e. On, B, (/)) -> ((rec(F, A)` B) = U.(rec(F, A)"B) <-> (rec(F, A)` if(B e. On, B, (/))) = U.(rec(F, A)"if(B e. On, B, (/)))))
7	2, 6	imbi12d 626	. . . 4 |- (B = if(B e. On, B, (/)) -> ((Lim B -> (rec(F, A)` B) = U.(rec(F, A)"B)) <-> (Lim if(B e. On, B, (/)) -> (rec(F, A)` if(B e. On, B, (/))) = U.(rec(F, A)"if(B e. On, B, (/))))))
8		0elon 3022	. . . . . 6 |- (/) e. On
9	8	elimel 2394	. . . . 5 |- if(B e. On, B, (/)) e. On
10	9	rdglim 3943	. . . 4 |- (Lim if(B e. On, B, (/)) -> (rec(F, A)` if(B e. On, B, (/))) = U.(rec(F, A)"if(B e. On, B, (/))))
11	7, 10	dedth 2383	. . 3 |- (B e. On -> (Lim B -> (rec(F, A)` B) = U.(rec(F, A)"B)))
12	11	imp 350	. 2 |- ((B e. On /\ Lim B) -> (rec(F, A)` B) = U.(rec(F, A)"B))
13	1, 12	sylancom 475	1 |- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U.(rec(F, A)"B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  (/)c0 2280  ifcif 2361  U.cuni 2503  Oncon0 2948  Lim wlim 2949  "cima 3173  ` cfv 3182  reccrdg 3931

This theorem is referenced by:  rdglim2 3949

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-if 2362  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-lim 2953  df-suc 2954  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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