Theorem seqrval 4664

Description: Value of the characteristic function of the inner recursion in df-seq 4661.

Hypotheses

Ref	Expression
seqrval.1	|- H = {<.z, w>. | w = <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>.}
seqrval.2	|- A e. V

Assertion

Ref	Expression
seqrval	|- (H` A) = <.((1st` A) + 1), ((2nd` A)S(F` ((1st` A) + 1)))>.

Distinct variable group(s):   z,A   z,S,w   z,F,w

Proof of Theorem seqrval

Step	Hyp	Ref	Expression
1		seqrval.1	. . 3 |- H = {<.z, w>. | w = <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>.}
2	1	fveq1i 2833	. 2 |- (H` A) = ({<.z, w>. | w = <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>.}` A)
3		seqrval.2	. . 3 |- A e. V
4		opex 1893	. . 3 |- <.((1st` A) + 1), ((2nd` A)S(F` ((1st` A) + 1)))>. e. V
5		opeq12 1878	. . . 4 |- ((((1st` z) + 1) = ((1st` A) + 1) /\ ((2nd` z)S(F` ((1st` z) + 1))) = ((2nd` A)S(F` ((1st` A) + 1)))) -> <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>. = <.((1st` A) + 1), ((2nd` A)S(F` ((1st` A) + 1)))>.)
6		fveq2 2832	. . . . 5 |- (z = A -> (1st` z) = (1st` A))
7	6	opreq1d 3012	. . . 4 |- (z = A -> ((1st` z) + 1) = ((1st` A) + 1))
8		fveq2 2832	. . . . 5 |- (z = A -> (2nd` z) = (2nd` A))
9	7	fveq2d 2836	. . . . 5 |- (z = A -> (F` ((1st` z) + 1)) = (F` ((1st` A) + 1)))
10	8, 9	opreq12d 3014	. . . 4 |- (z = A -> ((2nd` z)S(F` ((1st` z) + 1))) = ((2nd` A)S(F` ((1st` A) + 1))))
11	5, 7, 10	sylanc 361	. . 3 |- (z = A -> <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>. = <.((1st` A) + 1), ((2nd` A)S(F` ((1st` A) + 1)))>.)
12	3, 4, 11	fvopab 2877	. 2 |- ({<.z, w>. | w = <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>.}` A) = <.((1st` A) + 1), ((2nd` A)S(F` ((1st` A) + 1)))>.
13	2, 12	eqtr 1119	1 |- (H` A) = <.((1st` A) + 1), ((2nd` A)S(F` ((1st` A) + 1)))>.

Syntax hints:   = wceq 1091   e. wcel 1092  Vcvv 1348  <.cop 1810  {copab 2055  ` cfv 2422  (class class class)co 3001  1stc1st 3085  2ndc2nd 3086  1c1 4029   + caddc 4031

This theorem is referenced by:  seqsuclem 4669

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003

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