Theorem seq1rval 6248

Description: Value of the characteristic function of the inner recursion in df-seq1 6245.

Hypotheses

Ref	Expression
seq1rval.1	|- H = {<.z, w>. | w = <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>.}
seq1rval.2	|- A e. V

Assertion

Ref	Expression
seq1rval	|- (H` A) = <.((1st` A) + 1), ((2nd` A)S(F` ((1st` A) + 1)))>.

Distinct variable groups:   z,A   z,S,w   z,F,w

Proof of Theorem seq1rval

Step	Hyp	Ref	Expression
1		seq1rval.1	. . 3 |- H = {<.z, w>. | w = <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>.}
2	1	fveq1i 3710	. 2 |- (H` A) = ({<.z, w>. | w = <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>.}` A)
3		seq1rval.2	. . 3 |- A e. V
4		opex 2772	. . 3 |- <.((1st` A) + 1), ((2nd` A)S(F` ((1st` A) + 1)))>. e. V
5		fveq2 3709	. . . . 5 |- (z = A -> (1st` z) = (1st` A))
6	5	opreq1d 3960	. . . 4 |- (z = A -> ((1st` z) + 1) = ((1st` A) + 1))
7		fveq2 3709	. . . . 5 |- (z = A -> (2nd` z) = (2nd` A))
8	6	fveq2d 3713	. . . . 5 |- (z = A -> (F` ((1st` z) + 1)) = (F` ((1st` A) + 1)))
9	7, 8	opreq12d 3963	. . . 4 |- (z = A -> ((2nd` z)S(F` ((1st` z) + 1))) = ((2nd` A)S(F` ((1st` A) + 1))))
10	6, 9	opeq12d 2486	. . 3 |- (z = A -> <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>. = <.((1st` A) + 1), ((2nd` A)S(F` ((1st` A) + 1)))>.)
11	3, 4, 10	fvopab 3775	. 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)))>.
12	2, 11	eqtr 1487	1 |- (H` A) = <.((1st` A) + 1), ((2nd` A)S(F` ((1st` A) + 1)))>.

Syntax hints:   = wceq 953   e. wcel 955  Vcvv 1802  <.cop 2401  {copab 2656  ` cfv 3172  (class class class)co 3948  1stc1st 4061  2ndc2nd 4062  1c1 5207   + caddc 5209

This theorem is referenced by:  seq1suclem 6253

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-opr 3950

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