Theorem frsucmptn 7534

Description: The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class D is a proper class). This is a technical lemma that can be used together with frsucmpt 7533 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypotheses

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

Assertion

Ref	Expression
frsucmptn	|- ( -. D e. _V -> ( F ` suc B ) = (/) )

Proof of Theorem frsucmptn

Step	Hyp	Ref	Expression
1		frsucmpt.4	. . 3 |- F = ( rec ( ( x e. _V |-> C ) , A ) |` om )
2	1	fveq1i 6192	. 2 |- ( F ` suc B ) = ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B )
3		frfnom 7530	. . . . . 6 |- ( rec ( ( x e. _V |-> C ) , A ) |` om ) Fn om
4		fndm 5990	. . . . . 6 |- ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) Fn om -> dom ( rec ( ( x e. _V |-> C ) , A ) |` om ) = om )
5	3, 4	ax-mp 5	. . . . 5 |- dom ( rec ( ( x e. _V |-> C ) , A ) |` om ) = om
6	5	eleq2i 2693	. . . 4 |- ( suc B e. dom ( rec ( ( x e. _V |-> C ) , A ) |` om ) <-> suc B e. om )
7		peano2b 7081	. . . . 5 |- ( B e. om <-> suc B e. om )
8		frsuc 7532	. . . . . . . 8 |- ( B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = ( ( x e. _V |-> C ) ` ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` B ) ) )
9	1	fveq1i 6192	. . . . . . . . 9 |- ( F ` B ) = ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` B )
10	9	fveq2i 6194	. . . . . . . 8 |- ( ( x e. _V |-> C ) ` ( F ` B ) ) = ( ( x e. _V |-> C ) ` ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` B ) )
11	8, 10	syl6eqr 2674	. . . . . . 7 |- ( B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = ( ( x e. _V |-> C ) ` ( F ` B ) ) )
12		nfmpt1 4747	. . . . . . . . . . . 12 |- F/_ x ( x e. _V |-> C )
13		frsucmpt.1	. . . . . . . . . . . 12 |- F/_ x A
14	12, 13	nfrdg 7510	. . . . . . . . . . 11 |- F/_ x rec ( ( x e. _V |-> C ) , A )
15		nfcv 2764	. . . . . . . . . . 11 |- F/_ x om
16	14, 15	nfres 5398	. . . . . . . . . 10 |- F/_ x ( rec ( ( x e. _V |-> C ) , A ) |` om )
17	1, 16	nfcxfr 2762	. . . . . . . . 9 |- F/_ x F
18		frsucmpt.2	. . . . . . . . 9 |- F/_ x B
19	17, 18	nffv 6198	. . . . . . . 8 |- F/_ x ( F ` B )
20		frsucmpt.3	. . . . . . . 8 |- F/_ x D
21		frsucmpt.5	. . . . . . . 8 |- ( x = ( F ` B ) -> C = D )
22		eqid 2622	. . . . . . . 8 |- ( x e. _V |-> C ) = ( x e. _V |-> C )
23	19, 20, 21, 22	fvmptnf 6302	. . . . . . 7 |- ( -. D e. _V -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = (/) )
24	11, 23	sylan9eqr 2678	. . . . . 6 |- ( ( -. D e. _V /\ B e. om ) -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) )
25	24	ex 450	. . . . 5 |- ( -. D e. _V -> ( B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) ) )
26	7, 25	syl5bir 233	. . . 4 |- ( -. D e. _V -> ( suc B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) ) )
27	6, 26	syl5bi 232	. . 3 |- ( -. D e. _V -> ( suc B e. dom ( rec ( ( x e. _V |-> C ) , A ) |` om ) -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) ) )
28		ndmfv 6218	. . 3 |- ( -. suc B e. dom ( rec ( ( x e. _V |-> C ) , A ) |` om ) -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) )
29	27, 28	pm2.61d1 171	. 2 |- ( -. D e. _V -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) )
30	2, 29	syl5eq 2668	1 |- ( -. D e. _V -> ( F ` suc B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   |` cres 5116   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   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-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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  trpredlem1  31727