Theorem rdgsucmptf 7084

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 7080	. . 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 5858	. . 3 |- ( F ` suc B ) = ( rec ( ( x e. _V |-> C ) , A ) ` suc B )
4	2	fveq1i 5858	. . . 4 |- ( F ` B ) = ( rec ( ( x e. _V |-> C ) , A ) ` B )
5	4	fveq2i 5860	. . 3 |- ( ( x e. _V |-> C ) ` ( F ` B ) ) = ( ( x e. _V |-> C ) ` ( rec ( ( x e. _V |-> C ) , A ) ` B ) )
6	1, 3, 5	3eqtr4g 2526	. 2 |- ( B e. On -> ( F ` suc B ) = ( ( x e. _V |-> C ) ` ( F ` B ) ) )
7		fvex 5867	. . 3 |- ( F ` B ) e. _V
8		nfmpt1 4529	. . . . . . 7 |- F/_ x ( x e. _V |-> C )
9		rdgsucmptf.1	. . . . . . 7 |- F/_ x A
10	8, 9	nfrdg 7070	. . . . . 6 |- F/_ x rec ( ( x e. _V |-> C ) , A )
11	2, 10	nfcxfr 2620	. . . . 5 |- F/_ x F
12		rdgsucmptf.2	. . . . 5 |- F/_ x B
13	11, 12	nffv 5864	. . . 4 |- F/_ x ( F ` B )
14		rdgsucmptf.3	. . . 4 |- F/_ x D
15		rdgsucmptf.5	. . . 4 |- ( x = ( F ` B ) -> C = D )
16		eqid 2460	. . . 4 |- ( x e. _V |-> C ) = ( x e. _V |-> C )
17	13, 14, 15, 16	fvmptf 5957	. . 3 |- ( ( ( F ` B ) e. _V /\ D e. V ) -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = D )
18	7, 17	mpan 670	. 2 |- ( D e. V -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = D )
19	6, 18	sylan9eq 2521	1 |- ( ( B e. On /\ D e. V ) -> ( F ` suc B ) = D )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   F/_wnfc 2608   _Vcvv 3106   |-> cmpt 4498   Oncon0 4871   suc csuc 4873   ` cfv 5579   reccrdg 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-recs 7032  df-rdg 7066

This theorem is referenced by:  rdgsucmpt2  7086  rdgsucmpt  7087