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 ∈ 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 ∈ V
4		opex 1893	. . 3 ⊢ ⟨((1st ‘A) + 1), ((2nd ‘A)S(F ‘((1st ‘A) + 1)))⟩ ∈ 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   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810  {copab 2055   ‘cfv 2422  (class class class)co 3001  1^st c1st 3085  2^nd c2nd 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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