Theorem frsucmptn 7156

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 7155 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 5866	. 2 |- ( F ` suc B ) = ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B )
3		frfnom 7152	. . . . . 6 |- ( rec ( ( x e. _V |-> C ) , A ) |` om ) Fn om
4		fndm 5675	. . . . . 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 2521	. . . 4 |- ( suc B e. dom ( rec ( ( x e. _V |-> C ) , A ) |` om ) <-> suc B e. om )
7		peano2b 6708	. . . . 5 |- ( B e. om <-> suc B e. om )
8		frsuc 7154	. . . . . . . 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 5866	. . . . . . . . 9 |- ( F ` B ) = ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` B )
10	9	fveq2i 5868	. . . . . . . 8 |- ( ( x e. _V |-> C ) ` ( F ` B ) ) = ( ( x e. _V |-> C ) ` ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` B ) )
11	8, 10	syl6eqr 2503	. . . . . . 7 |- ( B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = ( ( x e. _V |-> C ) ` ( F ` B ) ) )
12		nfmpt1 4492	. . . . . . . . . . . 12 |- F/_ x ( x e. _V |-> C )
13		frsucmpt.1	. . . . . . . . . . . 12 |- F/_ x A
14	12, 13	nfrdg 7132	. . . . . . . . . . 11 |- F/_ x rec ( ( x e. _V |-> C ) , A )
15		nfcv 2592	. . . . . . . . . . 11 |- F/_ x om
16	14, 15	nfres 5107	. . . . . . . . . 10 |- F/_ x ( rec ( ( x e. _V |-> C ) , A ) |` om )
17	1, 16	nfcxfr 2590	. . . . . . . . 9 |- F/_ x F
18		frsucmpt.2	. . . . . . . . 9 |- F/_ x B
19	17, 18	nffv 5872	. . . . . . . 8 |- F/_ x ( F ` B )
20		frsucmpt.3	. . . . . . . 8 |- F/_ x D
21		frsucmpt.5	. . . . . . . 8 |- ( x = ( F ` B ) -> C = D )
22		eqid 2451	. . . . . . . 8 |- ( x e. _V |-> C ) = ( x e. _V |-> C )
23	19, 20, 21, 22	fvmptnf 5967	. . . . . . 7 |- ( -. D e. _V -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = (/) )
24	11, 23	sylan9eqr 2507	. . . . . 6 |- ( ( -. D e. _V /\ B e. om ) -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) )
25	24	ex 436	. . . . 5 |- ( -. D e. _V -> ( B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) ) )
26	7, 25	syl5bir 222	. . . 4 |- ( -. D e. _V -> ( suc B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) ) )
27	6, 26	syl5bi 221	. . 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 5889	. . 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 163	. 2 |- ( -. D e. _V -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) )
30	2, 29	syl5eq 2497	1 |- ( -. D e. _V -> ( F ` suc B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1444   e. wcel 1887   F/_wnfc 2579   _Vcvv 3045   (/)c0 3731   |-> cmpt 4461   dom cdm 4834   |` cres 4836   suc csuc 5425   Fn wfn 5577   ` cfv 5582   omcom 6692   reccrdg 7127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128

This theorem is referenced by:  trpredlem1  30468