Theorem r1sucg 7409

Description: Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
r1sucg	|- ( A e. dom R1 -> ( R1 ` suc A ) = ~P ( R1 ` A ) )

Proof of Theorem r1sucg

Step	Hyp	Ref	Expression
1		rdgsucg 6404	. . 3 |- ( A e. dom rec ( ( x e. _V |-> ~P x ) , (/) ) -> ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` suc A ) = ( ( x e. _V |-> ~P x ) ` ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A ) ) )
2		df-r1 7404	. . . 4 |- R1 = rec ( ( x e. _V |-> ~P x ) , (/) )
3	2	dmeqi 4868	. . 3 |- dom R1 = dom rec ( ( x e. _V |-> ~P x ) , (/) )
4	1, 3	eleq2s 2350	. 2 |- ( A e. dom R1 -> ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` suc A ) = ( ( x e. _V |-> ~P x ) ` ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A ) ) )
5	2	fveq1i 5459	. 2 |- ( R1 ` suc A ) = ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` suc A )
6		fvex 5472	. . . 4 |- ( R1 ` A ) e. _V
7		pweq 3602	. . . . 5 |- ( x = ( R1 ` A ) -> ~P x = ~P ( R1 ` A ) )
8		eqid 2258	. . . . 5 |- ( x e. _V |-> ~P x ) = ( x e. _V |-> ~P x )
9	6	pwex 4165	. . . . 5 |- ~P ( R1 ` A ) e. _V
10	7, 8, 9	fvmpt 5536	. . . 4 |- ( ( R1 ` A ) e. _V -> ( ( x e. _V |-> ~P x ) ` ( R1 ` A ) ) = ~P ( R1 ` A ) )
11	6, 10	ax-mp 10	. . 3 |- ( ( x e. _V |-> ~P x ) ` ( R1 ` A ) ) = ~P ( R1 ` A )
12	2	fveq1i 5459	. . . 4 |- ( R1 ` A ) = ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A )
13	12	fveq2i 5461	. . 3 |- ( ( x e. _V |-> ~P x ) ` ( R1 ` A ) ) = ( ( x e. _V |-> ~P x ) ` ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A ) )
14	11, 13	eqtr3i 2280	. 2 |- ~P ( R1 ` A ) = ( ( x e. _V |-> ~P x ) ` ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A ) )
15	4, 5, 14	3eqtr4g 2315	1 |- ( A e. dom R1 -> ( R1 ` suc A ) = ~P ( R1 ` A ) )

Syntax hints:   -> wi 6   = wceq 1619   e. wcel 1621   _Vcvv 2763   (/)c0 3430   ~Pcpw 3599   e. cmpt 4051   suc csuc 4366   dom cdm 4661   ` cfv 4673   reccrdg 6390   R1cr1 7402

This theorem is referenced by:  r1suc  7410  r1fin  7413  r1tr  7416  r1ordg  7418  r1pwss  7424  r1val1  7426  rankwflemb  7433  r1elwf  7436  rankr1ai  7438  rankr1bg  7443  pwwf  7447  unwf  7450  uniwf  7459  rankonidlem  7468  rankr1id  7502

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-recs 6356  df-rdg 6391  df-r1 7404