Theorem frsucmptn 6906

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 6905 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 5704	. 2 |- ( F ` suc B ) = ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B )
3		frfnom 6902	. . . . . 6 |- ( rec ( ( x e. _V |-> C ) , A ) |` om ) Fn om
4		fndm 5522	. . . . . 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 2507	. . . 4 |- ( suc B e. dom ( rec ( ( x e. _V |-> C ) , A ) |` om ) <-> suc B e. om )
7		peano2b 6504	. . . . 5 |- ( B e. om <-> suc B e. om )
8		frsuc 6904	. . . . . . . 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 5704	. . . . . . . . 9 |- ( F ` B ) = ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` B )
10	9	fveq2i 5706	. . . . . . . 8 |- ( ( x e. _V |-> C ) ` ( F ` B ) ) = ( ( x e. _V |-> C ) ` ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` B ) )
11	8, 10	syl6eqr 2493	. . . . . . 7 |- ( B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = ( ( x e. _V |-> C ) ` ( F ` B ) ) )
12		nfmpt1 4393	. . . . . . . . . . . 12 |- F/_ x ( x e. _V |-> C )
13		frsucmpt.1	. . . . . . . . . . . 12 |- F/_ x A
14	12, 13	nfrdg 6882	. . . . . . . . . . 11 |- F/_ x rec ( ( x e. _V |-> C ) , A )
15		nfcv 2589	. . . . . . . . . . 11 |- F/_ x om
16	14, 15	nfres 5124	. . . . . . . . . 10 |- F/_ x ( rec ( ( x e. _V |-> C ) , A ) |` om )
17	1, 16	nfcxfr 2587	. . . . . . . . 9 |- F/_ x F
18		frsucmpt.2	. . . . . . . . 9 |- F/_ x B
19	17, 18	nffv 5710	. . . . . . . 8 |- F/_ x ( F ` B )
20		frsucmpt.3	. . . . . . . 8 |- F/_ x D
21		frsucmpt.5	. . . . . . . 8 |- ( x = ( F ` B ) -> C = D )
22		eqid 2443	. . . . . . . 8 |- ( x e. _V |-> C ) = ( x e. _V |-> C )
23	19, 20, 21, 22	fvmptnf 5803	. . . . . . 7 |- ( -. D e. _V -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = (/) )
24	11, 23	sylan9eqr 2497	. . . . . 6 |- ( ( -. D e. _V /\ B e. om ) -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) )
25	24	ex 434	. . . . 5 |- ( -. D e. _V -> ( B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) ) )
26	7, 25	syl5bir 218	. . . 4 |- ( -. D e. _V -> ( suc B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) ) )
27	6, 26	syl5bi 217	. . 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 5726	. . 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 159	. 2 |- ( -. D e. _V -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) )
30	2, 29	syl5eq 2487	1 |- ( -. D e. _V -> ( F ` suc B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1369   e. wcel 1756   F/_wnfc 2575   _Vcvv 2984   (/)c0 3649   e. cmpt 4362   suc csuc 4733   dom cdm 4852   |` cres 4854   Fn wfn 5425   ` cfv 5430   omcom 6488   reccrdg 6877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-om 6489  df-recs 6844  df-rdg 6878

This theorem is referenced by:  trpredlem1  27703