Theorem frsucmptn 7105

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 7104 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 5867	. 2 |- ( F ` suc B ) = ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B )
3		frfnom 7101	. . . . . 6 |- ( rec ( ( x e. _V |-> C ) , A ) |` om ) Fn om
4		fndm 5680	. . . . . 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 2545	. . . 4 |- ( suc B e. dom ( rec ( ( x e. _V |-> C ) , A ) |` om ) <-> suc B e. om )
7		peano2b 6701	. . . . 5 |- ( B e. om <-> suc B e. om )
8		frsuc 7103	. . . . . . . 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 5867	. . . . . . . . 9 |- ( F ` B ) = ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` B )
10	9	fveq2i 5869	. . . . . . . 8 |- ( ( x e. _V |-> C ) ` ( F ` B ) ) = ( ( x e. _V |-> C ) ` ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` B ) )
11	8, 10	syl6eqr 2526	. . . . . . 7 |- ( B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = ( ( x e. _V |-> C ) ` ( F ` B ) ) )
12		nfmpt1 4536	. . . . . . . . . . . 12 |- F/_ x ( x e. _V |-> C )
13		frsucmpt.1	. . . . . . . . . . . 12 |- F/_ x A
14	12, 13	nfrdg 7081	. . . . . . . . . . 11 |- F/_ x rec ( ( x e. _V |-> C ) , A )
15		nfcv 2629	. . . . . . . . . . 11 |- F/_ x om
16	14, 15	nfres 5275	. . . . . . . . . 10 |- F/_ x ( rec ( ( x e. _V |-> C ) , A ) |` om )
17	1, 16	nfcxfr 2627	. . . . . . . . 9 |- F/_ x F
18		frsucmpt.2	. . . . . . . . 9 |- F/_ x B
19	17, 18	nffv 5873	. . . . . . . 8 |- F/_ x ( F ` B )
20		frsucmpt.3	. . . . . . . 8 |- F/_ x D
21		frsucmpt.5	. . . . . . . 8 |- ( x = ( F ` B ) -> C = D )
22		eqid 2467	. . . . . . . 8 |- ( x e. _V |-> C ) = ( x e. _V |-> C )
23	19, 20, 21, 22	fvmptnf 5968	. . . . . . 7 |- ( -. D e. _V -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = (/) )
24	11, 23	sylan9eqr 2530	. . . . . 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 5890	. . 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 2520	1 |- ( -. D e. _V -> ( F ` suc B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1379   e. wcel 1767   F/_wnfc 2615   _Vcvv 3113   (/)c0 3785   |-> cmpt 4505   suc csuc 4880   dom cdm 4999   |` cres 5001   Fn wfn 5583   ` cfv 5588   omcom 6685   reccrdg 7076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6686  df-recs 7043  df-rdg 7077

This theorem is referenced by:  trpredlem1  29163