Theorem r1val3 8701

Description: The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

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		r1fnon 8630	. . . . 5 |- R1 Fn On
2		fndm 5990	. . . . 5 |- ( R1 Fn On -> dom R1 = On )
3	1, 2	ax-mp 5	. . . 4 |- dom R1 = On
4	3	eleq2i 2693	. . 3 |- ( A e. dom R1 <-> A e. On )
5		r1val1 8649	. . 3 |- ( A e. dom R1 -> ( R1 ` A ) = U_ x e. A ~P ( R1 ` x ) )
6	4, 5	sylbir 225	. 2 |- ( A e. On -> ( R1 ` A ) = U_ x e. A ~P ( R1 ` x ) )
7		onelon 5748	. . . . 5 |- ( ( A e. On /\ x e. A ) -> x e. On )
8		r1val2 8700	. . . . 5 |- ( x e. On -> ( R1 ` x ) = { y | ( rank ` y ) e. x } )
9	7, 8	syl 17	. . . 4 |- ( ( A e. On /\ x e. A ) -> ( R1 ` x ) = { y | ( rank ` y ) e. x } )
10	9	pweqd 4163	. . 3 |- ( ( A e. On /\ x e. A ) -> ~P ( R1 ` x ) = ~P { y | ( rank ` y ) e. x } )
11	10	iuneq2dv 4542	. 2 |- ( A e. On -> U_ x e. A ~P ( R1 ` x ) = U_ x e. A ~P { y | ( rank ` y ) e. x } )
12	6, 11	eqtrd 2656	1 |- ( A e. On -> ( R1 ` A ) = U_ x e. A ~P { y | ( rank ` y ) e. x } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   ~Pcpw 4158   U_ciun 4520   dom cdm 5114   Oncon0 5723   Fn wfn 5883   ` cfv 5888   R1cr1 8625   rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by: (None)