Theorem rdgsucopabn 4005

Description: The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class D is a proper class). This is a technical lemma that can be used together with rdgsucopab 4004 to help eliminate redundant sethood antecedents.

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
rdgsucopabn	⊢ (¬ D ∈ V → (F ‘suc B) = ∅)

Distinct variable groups:   z,D   y,z,C   z,A   z,B   x,y,z

Proof of Theorem rdgsucopabn

Step	Hyp	Ref	Expression
1		rdgsuc 4003	. . . . 5 ⊢ (B ∈ On → (rec({⟨x, y⟩∣y = C}, A) ‘suc B) = ({⟨x, y⟩∣y = C} ‘(rec({⟨x, y⟩∣y = C}, A) ‘B)))
2		rdgsucopab.4	. . . . . 6 ⊢ F = rec({⟨x, y⟩∣y = C}, A)
3	2	fveq1i 3782	. . . . 5 ⊢ (F ‘suc B) = (rec({⟨x, y⟩∣y = C}, A) ‘suc B)
4	1, 3	syl5eq 1566	. . . 4 ⊢ (B ∈ On → (F ‘suc B) = ({⟨x, y⟩∣y = C} ‘(rec({⟨x, y⟩∣y = C}, A) ‘B)))
5		hbopab1 2869	. . . . . . 7 ⊢ (z ∈ {⟨x, y⟩∣y = C} → ∀x z ∈ {⟨x, y⟩∣y = C})
6		rdgsucopab.1	. . . . . . 7 ⊢ (z ∈ A → ∀x z ∈ A)
7	5, 6	hbrdg 3994	. . . . . 6 ⊢ (z ∈ rec({⟨x, y⟩∣y = C}, A) → ∀x z ∈ rec({⟨x, y⟩∣y = C}, A))
8		rdgsucopab.2	. . . . . 6 ⊢ (z ∈ B → ∀x z ∈ B)
9	7, 8	hbfv 3786	. . . . 5 ⊢ (z ∈ (rec({⟨x, y⟩∣y = C}, A) ‘B) → ∀x z ∈ (rec({⟨x, y⟩∣y = C}, A) ‘B))
10		rdgsucopab.3	. . . . 5 ⊢ (z ∈ D → ∀x z ∈ D)
11	2	fveq1i 3782	. . . . . . 7 ⊢ (F ‘B) = (rec({⟨x, y⟩∣y = C}, A) ‘B)
12	11	eqeq2i 1532	. . . . . 6 ⊢ (x = (F ‘B) ↔ x = (rec({⟨x, y⟩∣y = C}, A) ‘B))
13		rdgsucopab.5	. . . . . 6 ⊢ (x = (F ‘B) → C = D)
14	12, 13	sylbir 208	. . . . 5 ⊢ (x = (rec({⟨x, y⟩∣y = C}, A) ‘B) → C = D)
15	9, 10, 14	fvopabnf 3845	. . . 4 ⊢ (¬ D ∈ V → ({⟨x, y⟩∣y = C} ‘(rec({⟨x, y⟩∣y = C}, A) ‘B)) = ∅)
16	4, 15	sylan9eq 1574	. . 3 ⊢ ((B ∈ On ⋀ ¬ D ∈ V) → (F ‘suc B) = ∅)
17	16	ex 380	. 2 ⊢ (B ∈ On → (¬ D ∈ V → (F ‘suc B) = ∅))
18		sucelon 3125	. . . . . 6 ⊢ (B ∈ On ↔ suc B ∈ On)
19	2	dmeqi 3369	. . . . . . . 8 ⊢ dom F = dom rec({⟨x, y⟩∣y = C}, A)
20		rdgfnon 3997	. . . . . . . . 9 ⊢ rec({⟨x, y⟩∣y = C}, A) Fn On
21		fndm 3644	. . . . . . . . 9 ⊢ (rec({⟨x, y⟩∣y = C}, A) Fn On → dom rec({⟨x, y⟩∣y = C}, A) = On)
22	20, 21	ax-mp 7	. . . . . . . 8 ⊢ dom rec({⟨x, y⟩∣y = C}, A) = On
23	19, 22	eqtri 1542	. . . . . . 7 ⊢ dom F = On
24	23	eleq2i 1585	. . . . . 6 ⊢ (suc B ∈ dom F ↔ suc B ∈ On)
25	18, 24	bitr4i 183	. . . . 5 ⊢ (B ∈ On ↔ suc B ∈ dom F)
26	25	notbii 194	. . . 4 ⊢ (¬ B ∈ On ↔ ¬ suc B ∈ dom F)
27		ndmfv 3802	. . . 4 ⊢ (¬ suc B ∈ dom F → (F ‘suc B) = ∅)
28	26, 27	sylbi 206	. . 3 ⊢ (¬ B ∈ On → (F ‘suc B) = ∅)
29	28	a1d 12	. 2 ⊢ (¬ B ∈ On → (¬ D ∈ V → (F ‘suc B) = ∅))
30	17, 29	pm2.61i 132	1 ⊢ (¬ D ∈ V → (F ‘suc B) = ∅)

Syntax hints:  ¬ wn 2   → wi 3  ∀wal 995   = wceq 997   ∈ wcel 999  Vcvv 1858  ∅c0 2331  {copab 2721  Oncon0 3005  suc csuc 3007  dom cdm 3227   Fn wfn 3234   ‘cfv 3239  reccrdg 3989

This theorem is referenced by:  alephon 4930

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-rab 1699  df-v 1859  df-sbc 1989  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-iun 2622  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-id 2891  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-lim 3010  df-suc 3011  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-fv 3255  df-rdg 3990

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