Theorem frsucmptn 7096

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 7095 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 5849	. 2 |- ( F ` suc B ) = ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B )
3		frfnom 7092	. . . . . 6 |- ( rec ( ( x e. _V |-> C ) , A ) |` om ) Fn om
4		fndm 5662	. . . . . 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 2532	. . . 4 |- ( suc B e. dom ( rec ( ( x e. _V |-> C ) , A ) |` om ) <-> suc B e. om )
7		peano2b 6689	. . . . 5 |- ( B e. om <-> suc B e. om )
8		frsuc 7094	. . . . . . . 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 5849	. . . . . . . . 9 |- ( F ` B ) = ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` B )
10	9	fveq2i 5851	. . . . . . . 8 |- ( ( x e. _V |-> C ) ` ( F ` B ) ) = ( ( x e. _V |-> C ) ` ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` B ) )
11	8, 10	syl6eqr 2513	. . . . . . 7 |- ( B e. om -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = ( ( x e. _V |-> C ) ` ( F ` B ) ) )
12		nfmpt1 4528	. . . . . . . . . . . 12 |- F/_ x ( x e. _V |-> C )
13		frsucmpt.1	. . . . . . . . . . . 12 |- F/_ x A
14	12, 13	nfrdg 7072	. . . . . . . . . . 11 |- F/_ x rec ( ( x e. _V |-> C ) , A )
15		nfcv 2616	. . . . . . . . . . 11 |- F/_ x om
16	14, 15	nfres 5264	. . . . . . . . . 10 |- F/_ x ( rec ( ( x e. _V |-> C ) , A ) |` om )
17	1, 16	nfcxfr 2614	. . . . . . . . 9 |- F/_ x F
18		frsucmpt.2	. . . . . . . . 9 |- F/_ x B
19	17, 18	nffv 5855	. . . . . . . 8 |- F/_ x ( F ` B )
20		frsucmpt.3	. . . . . . . 8 |- F/_ x D
21		frsucmpt.5	. . . . . . . 8 |- ( x = ( F ` B ) -> C = D )
22		eqid 2454	. . . . . . . 8 |- ( x e. _V |-> C ) = ( x e. _V |-> C )
23	19, 20, 21, 22	fvmptnf 5949	. . . . . . 7 |- ( -. D e. _V -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = (/) )
24	11, 23	sylan9eqr 2517	. . . . . 6 |- ( ( -. D e. _V /\ B e. om ) -> ( ( rec ( ( x e. _V |-> C ) , A ) |` om ) ` suc B ) = (/) )
25	24	ex 432	. . . . 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 5872	. . 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 2507	1 |- ( -. D e. _V -> ( F ` suc B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   e. wcel 1823   F/_wnfc 2602   _Vcvv 3106   (/)c0 3783   |-> cmpt 4497   suc csuc 4869   dom cdm 4988   |` cres 4990   Fn wfn 5565   ` cfv 5570   omcom 6673   reccrdg 7067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068

This theorem is referenced by:  trpredlem1  29550