Theorem frsucmpt2 7535

Description: The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation), using double-substitution instead of a bound variable condition. (Contributed by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
frsucmpt2.1	|- F = ( rec ( ( x e. _V |-> C ) , A ) |` om )
frsucmpt2.2	|- ( y = x -> E = C )
frsucmpt2.3	|- ( y = ( F ` B ) -> E = D )

Assertion

Ref	Expression
frsucmpt2	|- ( ( B e. om /\ D e. V ) -> ( F ` suc B ) = D )

Distinct variable groups:   y, A   y, B   y, C   y, D   x, E

Allowed substitution hints:   A( x)   B( x)   C( x)   D( x)   E( y)   F( x, y)   V( x, y)

Proof of Theorem frsucmpt2

Step	Hyp	Ref	Expression
1		nfcv 2764	. 2 |- F/_ y A
2		nfcv 2764	. 2 |- F/_ y B
3		nfcv 2764	. 2 |- F/_ y D
4		frsucmpt2.1	. . 3 |- F = ( rec ( ( x e. _V |-> C ) , A ) |` om )
5		frsucmpt2.2	. . . . . 6 |- ( y = x -> E = C )
6	5	cbvmptv 4750	. . . . 5 |- ( y e. _V |-> E ) = ( x e. _V |-> C )
7		rdgeq1 7507	. . . . 5 |- ( ( y e. _V |-> E ) = ( x e. _V |-> C ) -> rec ( ( y e. _V |-> E ) , A ) = rec ( ( x e. _V |-> C ) , A ) )
8	6, 7	ax-mp 5	. . . 4 |- rec ( ( y e. _V |-> E ) , A ) = rec ( ( x e. _V |-> C ) , A )
9	8	reseq1i 5392	. . 3 |- ( rec ( ( y e. _V |-> E ) , A ) |` om ) = ( rec ( ( x e. _V |-> C ) , A ) |` om )
10	4, 9	eqtr4i 2647	. 2 |- F = ( rec ( ( y e. _V |-> E ) , A ) |` om )
11		frsucmpt2.3	. 2 |- ( y = ( F ` B ) -> E = D )
12	1, 2, 3, 10, 11	frsucmpt 7533	1 |- ( ( B e. om /\ D e. V ) -> ( F ` suc B ) = D )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   |` cres 5116   suc csuc 5725   ` 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:  unblem2  8213  unblem3  8214  inf0  8518  trcl  8604  hsmexlem8  9246  wunex2  9560  wuncval2  9569  peano5nni  11023  peano2nn  11032  om2uzsuci  12747  neibastop2lem  32355