Theorem rdgsucopabn 3942

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 3941 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		rdgsuct 3940	. . . . 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 3720	. . . . 5 ⊢ (F ‘suc B) = (rec({⟨x, y⟩∣y = C}, A) ‘suc B)
4	1, 3	syl5eq 1517	. . . 4 ⊢ (B ∈ On → (F ‘suc B) = ({⟨x, y⟩∣y = C} ‘(rec({⟨x, y⟩∣y = C}, A) ‘B)))
5		hbopab1 2809	. . . . . . 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 3931	. . . . . 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 3724	. . . . 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 3720	. . . . . . 7 ⊢ (F ‘B) = (rec({⟨x, y⟩∣y = C}, A) ‘B)
12	11	eqeq2i 1483	. . . . . 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 201	. . . . 5 ⊢ (x = (rec({⟨x, y⟩∣y = C}, A) ‘B) → C = D)
15	9, 10, 14	fvopabnf 3783	. . . 4 ⊢ (¬ D ∈ V → ({⟨x, y⟩∣y = C} ‘(rec({⟨x, y⟩∣y = C}, A) ‘B)) = ∅)
16	4, 15	sylan9eq 1525	. . 3 ⊢ ((B ∈ On ⋀ ¬ D ∈ V) → (F ‘suc B) = ∅)
17	16	ex 373	. 2 ⊢ (B ∈ On → (¬ D ∈ V → (F ‘suc B) = ∅))
18		sucelon 3064	. . . . . 6 ⊢ (B ∈ On ↔ suc B ∈ On)
19	2	dmeqi 3308	. . . . . . . 8 ⊢ dom F = dom rec({⟨x, y⟩∣y = C}, A)
20		rdgfnon 3934	. . . . . . . . 9 ⊢ rec({⟨x, y⟩∣y = C}, A) Fn On
21		fndm 3583	. . . . . . . . 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	eqtr 1493	. . . . . . 7 ⊢ dom F = On
24	23	eleq2i 1536	. . . . . 6 ⊢ (suc B ∈ dom F ↔ suc B ∈ On)
25	18, 24	bitr4 176	. . . . 5 ⊢ (B ∈ On ↔ suc B ∈ dom F)
26	25	negbii 187	. . . 4 ⊢ (¬ B ∈ On ↔ ¬ suc B ∈ dom F)
27		ndmfv 3740	. . . 4 ⊢ (¬ suc B ∈ dom F → (F ‘suc B) = ∅)
28	26, 27	sylbi 199	. . 3 ⊢ (¬ B ∈ On → (F ‘suc B) = ∅)
29	28	a1d 12	. 2 ⊢ (¬ B ∈ On → (¬ D ∈ V → (F ‘suc B) = ∅))
30	17, 29	pm2.61i 126	1 ⊢ (¬ D ∈ V → (F ‘suc B) = ∅)

Syntax hints:  ¬ wn 2   → wi 3  ∀wal 953   = wceq 955   ∈ wcel 957  Vcvv 1808  ∅c0 2277  {copab 2662  Oncon0 2944  suc csuc 2946  dom cdm 3166   Fn wfn 3173   ‘cfv 3178  reccrdg 3926

This theorem is referenced by:  alephon 4848

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194  df-rdg 3927

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