Theorem seqval 12812

Description: Value of the sequence builder function. (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypothesis

Ref	Expression
seqval.1	|- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. ) |` om )

Assertion

Ref	Expression
seqval	|- seq M ( .+ , F ) = ran R

Distinct variable groups:   w, F, x, y, z   w, .+ , x, y, z   x, M, y

Allowed substitution hints:   R( x, y, z, w)   M( z, w)

Proof of Theorem seqval

Step	Hyp	Ref	Expression
1		df-ima 5127	. 2 |- ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) " om ) = ran ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) |` om )
2		df-seq 12802	. 2 |- seq M ( .+ , F ) = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) " om )
3		seqval.1	. . . 4 |- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. ) |` om )
4		eqid 2622	. . . . . . 7 |- _V = _V
5		vex 3203	. . . . . . . . 9 |- x e. _V
6		vex 3203	. . . . . . . . 9 |- y e. _V
7		oveq1 6657	. . . . . . . . . . . 12 |- ( z = x -> ( z + 1 ) = ( x + 1 ) )
8	7	fveq2d 6195	. . . . . . . . . . 11 |- ( z = x -> ( F ` ( z + 1 ) ) = ( F ` ( x + 1 ) ) )
9	8	oveq2d 6666	. . . . . . . . . 10 |- ( z = x -> ( w .+ ( F ` ( z + 1 ) ) ) = ( w .+ ( F ` ( x + 1 ) ) ) )
10		oveq1 6657	. . . . . . . . . 10 |- ( w = y -> ( w .+ ( F ` ( x + 1 ) ) ) = ( y .+ ( F ` ( x + 1 ) ) ) )
11		eqid 2622	. . . . . . . . . 10 |- ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) = ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) )
12		ovex 6678	. . . . . . . . . 10 |- ( y .+ ( F ` ( x + 1 ) ) ) e. _V
13	9, 10, 11, 12	ovmpt2 6796	. . . . . . . . 9 |- ( ( x e. _V /\ y e. _V ) -> ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) = ( y .+ ( F ` ( x + 1 ) ) ) )
14	5, 6, 13	mp2an 708	. . . . . . . 8 |- ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) = ( y .+ ( F ` ( x + 1 ) ) )
15	14	opeq2i 4406	. . . . . . 7 |- <. ( x + 1 ) , ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. = <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >.
16	4, 4, 15	mpt2eq123i 6718	. . . . . 6 |- ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) = ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. )
17		rdgeq1 7507	. . . . . 6 |- ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) = ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) -> rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. ) = rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
18	16, 17	ax-mp 5	. . . . 5 |- rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. ) = rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
19	18	reseq1i 5392	. . . 4 |- ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x ( z e. _V , w e. _V |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. ) |` om ) = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) |` om )
20	3, 19	eqtri 2644	. . 3 |- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) |` om )
21	20	rneqi 5352	. 2 |- ran R = ran ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) |` om )
22	1, 2, 21	3eqtr4i 2654	1 |- seq M ( .+ , F ) = ran R

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   ran crn 5115   |` cres 5116   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   reccrdg 7505   1c1 9937   + caddc 9939   seqcseq 12801

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802

This theorem is referenced by:  seqfn  12813  seq1  12814  seqp1  12816