Theorem r1sucg 7374

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 6369	. . 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 7369	. . . 4 |- R1 = rec ( ( x e. _V |-> ~P x ) , (/) )
3	2	dmeqi 4833	. . 3 |- dom R1 = dom rec ( ( x e. _V |-> ~P x ) , (/) )
4	1, 3	eleq2s 2348	. 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 5424	. 2 |- ( R1 ` suc A ) = ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` suc A )
6		fvex 5437	. . . 4 |- ( R1 ` A ) e. _V
7		pweq 3569	. . . . 5 |- ( x = ( R1 ` A ) -> ~P x = ~P ( R1 ` A ) )
8		eqid 2256	. . . . 5 |- ( x e. _V |-> ~P x ) = ( x e. _V |-> ~P x )
9	6	pwex 4131	. . . . 5 |- ~P ( R1 ` A ) e. _V
10	7, 8, 9	fvmpt 5501	. . . 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 5424	. . . 4 |- ( R1 ` A ) = ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A )
13	12	fveq2i 5426	. . 3 |- ( ( x e. _V |-> ~P x ) ` ( R1 ` A ) ) = ( ( x e. _V |-> ~P x ) ` ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A ) )
14	11, 13	eqtr3i 2278	. 2 |- ~P ( R1 ` A ) = ( ( x e. _V |-> ~P x ) ` ( rec ( ( x e. _V |-> ~P x ) , (/) ) ` A ) )
15	4, 5, 14	3eqtr4g 2313	1 |- ( A e. dom R1 -> ( R1 ` suc A ) = ~P ( R1 ` A ) )

Syntax hints:   -> wi 6   = wceq 1619   e. wcel 1621   _Vcvv 2740   (/)c0 3397   ~Pcpw 3566   e. cmpt 4017   suc csuc 4331   dom cdm 4626   ` cfv 4638   reccrdg 6355   R1cr1 7367

This theorem is referenced by:  r1suc  7375  r1fin  7378  r1tr  7381  r1ordg  7383  r1pwss  7389  r1val1  7391  rankwflemb  7398  r1elwf  7401  rankr1ai  7403  rankr1bg  7408  pwwf  7412  unwf  7415  uniwf  7424  rankonidlem  7433  rankr1id  7467

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 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449

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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-recs 6321  df-rdg 6356  df-r1 7369