Theorem rdgsucmptf 7131

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 7127	. . 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 5850	. . 3 |- ( F ` suc B ) = ( rec ( ( x e. _V |-> C ) , A ) ` suc B )
4	2	fveq1i 5850	. . . 4 |- ( F ` B ) = ( rec ( ( x e. _V |-> C ) , A ) ` B )
5	4	fveq2i 5852	. . 3 |- ( ( x e. _V |-> C ) ` ( F ` B ) ) = ( ( x e. _V |-> C ) ` ( rec ( ( x e. _V |-> C ) , A ) ` B ) )
6	1, 3, 5	3eqtr4g 2468	. 2 |- ( B e. On -> ( F ` suc B ) = ( ( x e. _V |-> C ) ` ( F ` B ) ) )
7		fvex 5859	. . 3 |- ( F ` B ) e. _V
8		nfmpt1 4484	. . . . . . 7 |- F/_ x ( x e. _V |-> C )
9		rdgsucmptf.1	. . . . . . 7 |- F/_ x A
10	8, 9	nfrdg 7117	. . . . . 6 |- F/_ x rec ( ( x e. _V |-> C ) , A )
11	2, 10	nfcxfr 2562	. . . . 5 |- F/_ x F
12		rdgsucmptf.2	. . . . 5 |- F/_ x B
13	11, 12	nffv 5856	. . . 4 |- F/_ x ( F ` B )
14		rdgsucmptf.3	. . . 4 |- F/_ x D
15		rdgsucmptf.5	. . . 4 |- ( x = ( F ` B ) -> C = D )
16		eqid 2402	. . . 4 |- ( x e. _V |-> C ) = ( x e. _V |-> C )
17	13, 14, 15, 16	fvmptf 5950	. . 3 |- ( ( ( F ` B ) e. _V /\ D e. V ) -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = D )
18	7, 17	mpan 668	. 2 |- ( D e. V -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = D )
19	6, 18	sylan9eq 2463	1 |- ( ( B e. On /\ D e. V ) -> ( F ` suc B ) = D )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1405   e. wcel 1842   F/_wnfc 2550   _Vcvv 3059   |-> cmpt 4453   Oncon0 5410   suc csuc 5412   ` cfv 5569   reccrdg 7112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-wrecs 7013  df-recs 7075  df-rdg 7113

This theorem is referenced by:  rdgsucmpt2  7133  rdgsucmpt  7134