Theorem rdgsucopab 2984

Description: The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered pair abstraction).

Hypotheses

Ref	Expression
rdgsucopab.1	⊢ (z ∈ A → ∀x z ∈ A)
rdgsucopab.2	⊢ (z ∈ B → ∀x z ∈ B)
rdgsucopab.3	⊢ (z ∈ D → ∀x z ∈ D)
rdgsucopab.4	⊢ F = rec({⟨x, y⟩∣y = C}, A)
rdgsucopab.5	⊢ (x = (F ‘B) → C = D)

Assertion

Ref	Expression
rdgsucopab	⊢ ((B ∈ On ∧ D ∈ R) → (F ‘suc B) = D)

Distinct variable group(s):   z,D   y,z,C   z,A   z,B   x,y,z

Proof of Theorem rdgsucopab

Step	Hyp	Ref	Expression
1		rdgsuct 2983	. . 3 ⊢ (B ∈ On → (rec({⟨x, y⟩∣y = C}, A) ‘suc B) = ({⟨x, y⟩∣y = C} ‘(rec({⟨x, y⟩∣y = C}, A) ‘B)))
2		rdgsucopab.4	. . . 4 ⊢ F = rec({⟨x, y⟩∣y = C}, A)
3	2	fveq1i 2833	. . 3 ⊢ (F ‘suc B) = (rec({⟨x, y⟩∣y = C}, A) ‘suc B)
4	1, 3	syl5eq 1136	. 2 ⊢ (B ∈ On → (F ‘suc B) = ({⟨x, y⟩∣y = C} ‘(rec({⟨x, y⟩∣y = C}, A) ‘B)))
5		fvex 2838	. . 3 ⊢ (rec({⟨x, y⟩∣y = C}, A) ‘B) ∈ V
6		hbopab1 2112	. . . . . 6 ⊢ (z ∈ {⟨x, y⟩∣y = C} → ∀x z ∈ {⟨x, y⟩∣y = C})
7		rdgsucopab.1	. . . . . 6 ⊢ (z ∈ A → ∀x z ∈ A)
8	6, 7	hbrdg 2974	. . . . 5 ⊢ (z ∈ rec({⟨x, y⟩∣y = C}, A) → ∀x z ∈ rec({⟨x, y⟩∣y = C}, A))
9		rdgsucopab.2	. . . . 5 ⊢ (z ∈ B → ∀x z ∈ B)
10	8, 9	hbfv 2837	. . . 4 ⊢ (z ∈ (rec({⟨x, y⟩∣y = C}, A) ‘B) → ∀x z ∈ (rec({⟨x, y⟩∣y = C}, A) ‘B))
11		rdgsucopab.3	. . . 4 ⊢ (z ∈ D → ∀x z ∈ D)
12	2	fveq1i 2833	. . . . . 6 ⊢ (F ‘B) = (rec({⟨x, y⟩∣y = C}, A) ‘B)
13	12	cleq2i 1111	. . . . 5 ⊢ (x = (F ‘B) ↔ x = (rec({⟨x, y⟩∣y = C}, A) ‘B))
14		rdgsucopab.5	. . . . 5 ⊢ (x = (F ‘B) → C = D)
15	13, 14	sylbir 176	. . . 4 ⊢ (x = (rec({⟨x, y⟩∣y = C}, A) ‘B) → C = D)
16	10, 11, 15	fvopabgf 2874	. . 3 ⊢ (((rec({⟨x, y⟩∣y = C}, A) ‘B) ∈ V ∧ D ∈ R) → ({⟨x, y⟩∣y = C} ‘(rec({⟨x, y⟩∣y = C}, A) ‘B)) = D)
17	5, 16	mpan 518	. 2 ⊢ (D ∈ R → ({⟨x, y⟩∣y = C} ‘(rec({⟨x, y⟩∣y = C}, A) ‘B)) = D)
18	4, 17	sylan9eq 1144	1 ⊢ ((B ∈ On ∧ D ∈ R) → (F ‘suc B) = D)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {copab 2055  Oncon0 2199  suc csuc 2201   ‘cfv 2422  reccrdg 2969

This theorem is referenced by:  abianfplem 2999  r1suc 3496  alephon 3671  alephsuc 3672

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  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-fn 2433  df-fv 2438  df-rdg 2970

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