Theorem r1lim 4653

Description: Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76.

Assertion

Ref	Expression
r1lim	|- ((A e. B /\ Lim A) -> (R1` A) = U_x e. A (R1` x))

Distinct variable group:   x,A

Proof of Theorem r1lim

Step	Hyp	Ref	Expression
1		rdglim2a 3950	. 2 |- ((A e. B /\ Lim A) -> (rec({<.y, z>. | z = P~y}, (/))` A) = U_x e. A (rec({<.y, z>. | z = P~y}, (/))` x))
2		df-r1 4643	. . 3 |- R1 = rec({<.y, z>. | z = P~y}, (/))
3	2	fveq1i 3725	. 2 |- (R1` A) = (rec({<.y, z>. | z = P~y}, (/))` A)
4	2	fveq1i 3725	. . . 4 |- (R1` x) = (rec({<.y, z>. | z = P~y}, (/))` x)
5	4	a1i 8	. . 3 |- (x e. A -> (R1` x) = (rec({<.y, z>. | z = P~y}, (/))` x))
6	5	iuneq2i 2580	. 2 |- U_x e. A (R1` x) = U_x e. A (rec({<.y, z>. | z = P~y}, (/))` x)
7	1, 3, 6	3eqtr4g 1531	1 |- ((A e. B /\ Lim A) -> (R1` A) = U_x e. A (R1` x))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  (/)c0 2280  P~cpw 2401  U_ciun 2566  {copab 2666  Lim wlim 2949  ` cfv 3182  reccrdg 3931  R1cr1 4641

This theorem is referenced by:  r1tr 4654  r1ord 4655  r1val1 4658  rankr1 4674  r1pwcl 4687

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  df-r1 4643

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