Theorem rdgsucmptf 7524

Description: The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
rdgsucmptf.1	|- F/_ x A
rdgsucmptf.2	|- F/_ x B
rdgsucmptf.3	|- F/_ x D
rdgsucmptf.4	|- F = rec ( ( x e. _V |-> C ) , A )
rdgsucmptf.5	|- ( x = ( F ` B ) -> C = D )

Assertion

Ref	Expression
rdgsucmptf	|- ( ( B e. On /\ D e. V ) -> ( F ` suc B ) = D )

Proof of Theorem rdgsucmptf

Step	Hyp	Ref	Expression
1		rdgsuc 7520	. . 3 |- ( B e. On -> ( rec ( ( x e. _V |-> C ) , A ) ` suc B ) = ( ( x e. _V |-> C ) ` ( rec ( ( x e. _V |-> C ) , A ) ` B ) ) )
2		rdgsucmptf.4	. . . 4 |- F = rec ( ( x e. _V |-> C ) , A )
3	2	fveq1i 6192	. . 3 |- ( F ` suc B ) = ( rec ( ( x e. _V |-> C ) , A ) ` suc B )
4	2	fveq1i 6192	. . . 4 |- ( F ` B ) = ( rec ( ( x e. _V |-> C ) , A ) ` B )
5	4	fveq2i 6194	. . 3 |- ( ( x e. _V |-> C ) ` ( F ` B ) ) = ( ( x e. _V |-> C ) ` ( rec ( ( x e. _V |-> C ) , A ) ` B ) )
6	1, 3, 5	3eqtr4g 2681	. 2 |- ( B e. On -> ( F ` suc B ) = ( ( x e. _V |-> C ) ` ( F ` B ) ) )
7		fvex 6201	. . 3 |- ( F ` B ) e. _V
8		nfmpt1 4747	. . . . . . 7 |- F/_ x ( x e. _V |-> C )
9		rdgsucmptf.1	. . . . . . 7 |- F/_ x A
10	8, 9	nfrdg 7510	. . . . . 6 |- F/_ x rec ( ( x e. _V |-> C ) , A )
11	2, 10	nfcxfr 2762	. . . . 5 |- F/_ x F
12		rdgsucmptf.2	. . . . 5 |- F/_ x B
13	11, 12	nffv 6198	. . . 4 |- F/_ x ( F ` B )
14		rdgsucmptf.3	. . . 4 |- F/_ x D
15		rdgsucmptf.5	. . . 4 |- ( x = ( F ` B ) -> C = D )
16		eqid 2622	. . . 4 |- ( x e. _V |-> C ) = ( x e. _V |-> C )
17	13, 14, 15, 16	fvmptf 6301	. . 3 |- ( ( ( F ` B ) e. _V /\ D e. V ) -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = D )
18	7, 17	mpan 706	. 2 |- ( D e. V -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = D )
19	6, 18	sylan9eq 2676	1 |- ( ( B e. On /\ D e. V ) -> ( F ` suc B ) = D )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   |-> cmpt 4729   Oncon0 5723   suc csuc 5725   ` cfv 5888   reccrdg 7505

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

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-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-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  rdgsucmpt2  7526  rdgsucmpt  7527