Theorem r1sucg 7437

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 6432	. . 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 7432	. . . 4 |- R1 = rec ( ( x e. _V |-> ~P x ) , (/) )
3	2	dmeqi 4879	. . 3 |- dom R1 = dom rec ( ( x e. _V |-> ~P x ) , (/) )
4	1, 3	eleq2s 2376	. 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 5487	. 2 |- ( R1 ` suc A ) = ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` suc A )
6		fvex 5500	. . . 4 |- ( R1 ` A ) e. _V
7		pweq 3629	. . . . 5 |- ( x = ( R1 ` A ) -> ~P x = ~P ( R1 ` A ) )
8		eqid 2284	. . . . 5 |- ( x e. _V |-> ~P x ) = ( x e. _V |-> ~P x )
9	6	pwex 4192	. . . . 5 |- ~P ( R1 ` A ) e. _V
10	7, 8, 9	fvmpt 5564	. . . 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 5487	. . . 4 |- ( R1 ` A ) = ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A )
13	12	fveq2i 5489	. . 3 |- ( ( x e. _V |-> ~P x ) ` ( R1 ` A ) ) = ( ( x e. _V |-> ~P x ) ` ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A ) )
14	11, 13	eqtr3i 2306	. 2 |- ~P ( R1 ` A ) = ( ( x e. _V |-> ~P x ) ` ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A ) )
15	4, 5, 14	3eqtr4g 2341	1 |- ( A e. dom R1 -> ( R1 ` suc A ) = ~P ( R1 ` A ) )

Syntax hints:   -> wi 4   = wceq 1623   e. wcel 1685   _Vcvv 2789   (/)c0 3456   ~Pcpw 3626   e. cmpt 4078   suc csuc 4393   dom cdm 4688   ` cfv 5221   reccrdg 6418   R1cr1 7430

This theorem is referenced by:  r1suc  7438  r1fin  7441  r1tr  7444  r1ordg  7446  r1pwss  7452  r1val1  7454  rankwflemb  7461  r1elwf  7464  rankr1ai  7466  rankr1bg  7471  pwwf  7475  unwf  7478  uniwf  7487  rankonidlem  7496  rankr1id  7530

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-rdg 6419  df-r1 7432