Theorem r1sucg 7655

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

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rdgsucg 6644	. . 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 7650	. . . 4 |- R1 = rec ( ( x e. _V |-> ~P x ) , (/) )
3	2	dmeqi 5034	. . 3 |- dom R1 = dom rec ( ( x e. _V |-> ~P x ) , (/) )
4	1, 3	eleq2s 2500	. 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 5692	. 2 |- ( R1 ` suc A ) = ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` suc A )
6		fvex 5705	. . . 4 |- ( R1 ` A ) e. _V
7		pweq 3766	. . . . 5 |- ( x = ( R1 ` A ) -> ~P x = ~P ( R1 ` A ) )
8		eqid 2408	. . . . 5 |- ( x e. _V |-> ~P x ) = ( x e. _V |-> ~P x )
9	6	pwex 4346	. . . . 5 |- ~P ( R1 ` A ) e. _V
10	7, 8, 9	fvmpt 5769	. . . 4 |- ( ( R1 ` A ) e. _V -> ( ( x e. _V |-> ~P x ) ` ( R1 ` A ) ) = ~P ( R1 ` A ) )
11	6, 10	ax-mp 8	. . 3 |- ( ( x e. _V |-> ~P x ) ` ( R1 ` A ) ) = ~P ( R1 ` A )
12	2	fveq1i 5692	. . . 4 |- ( R1 ` A ) = ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A )
13	12	fveq2i 5694	. . 3 |- ( ( x e. _V |-> ~P x ) ` ( R1 ` A ) ) = ( ( x e. _V |-> ~P x ) ` ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A ) )
14	11, 13	eqtr3i 2430	. 2 |- ~P ( R1 ` A ) = ( ( x e. _V |-> ~P x ) ` ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A ) )
15	4, 5, 14	3eqtr4g 2465	1 |- ( A e. dom R1 -> ( R1 ` suc A ) = ~P ( R1 ` A ) )

Syntax hints:   -> wi 4   = wceq 1649   e. wcel 1721   _Vcvv 2920   (/)c0 3592   ~Pcpw 3763   e. cmpt 4230   suc csuc 4547   dom cdm 4841   ` cfv 5417   reccrdg 6630   R1cr1 7648

This theorem is referenced by:  r1suc  7656  r1fin  7659  r1tr  7662  r1ordg  7664  r1pwss  7670  r1val1  7672  rankwflemb  7679  r1elwf  7682  rankr1ai  7684  rankr1bg  7689  pwwf  7693  unwf  7696  uniwf  7705  rankonidlem  7714  rankr1id  7748

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-recs 6596  df-rdg 6631  df-r1 7650