Theorem r1val3 4679

Description: The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113.

Assertion

Ref	Expression
r1val3	|- (A e. On -> (R1` A) = U_x e. A P~{y | (rank` y) e. x})

Distinct variable group:   x,y,A

Proof of Theorem r1val3

Step	Hyp	Ref	Expression
1		r1val1 4658	. 2 |- (A e. On -> (R1` A) = U_x e. A P~(R1` x))
2		onelon 2972	. . . . 5 |- ((A e. On /\ x e. A) -> x e. On)
3		r1val2 4678	. . . . 5 |- (x e. On -> (R1` x) = {y | (rank` y) e. x})
4		pweq 2403	. . . . 5 |- ((R1` x) = {y | (rank` y) e. x} -> P~(R1` x) = P~{y | (rank` y) e. x})
5	2, 3, 4	3syl 20	. . . 4 |- ((A e. On /\ x e. A) -> P~(R1` x) = P~{y | (rank` y) e. x})
6	5	r19.21aiva 1714	. . 3 |- (A e. On -> A.x e. A P~(R1` x) = P~{y | (rank` y) e. x})
7		iuneq2 2578	. . 3 |- (A.x e. A P~(R1` x) = P~{y | (rank` y) e. x} -> U_x e. A P~(R1` x) = U_x e. A P~{y | (rank` y) e. x})
8	6, 7	syl 10	. 2 |- (A e. On -> U_x e. A P~(R1` x) = U_x e. A P~{y | (rank` y) e. x})
9	1, 8	eqtrd 1507	1 |- (A e. On -> (R1` A) = U_x e. A P~{y | (rank` y) e. x})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  P~cpw 2401  U_ciun 2566  Oncon0 2948  ` cfv 3182  R1cr1 4641  rankcrnk 4642

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  ax-reg 4593  ax-inf2 4625

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-int 2534  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-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  df-r1 4643  df-rank 4644

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