Theorem rdgsucmptnf 8469

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 rdgsucmptf 8468 to help eliminate redundant sethood antecedents. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
rdgsucmptf.1	⊢ Ⅎ𝑥𝐴
rdgsucmptf.2	⊢ Ⅎ𝑥𝐵
rdgsucmptf.3	⊢ Ⅎ𝑥𝐷
rdgsucmptf.4	⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
rdgsucmptf.5	⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
rdgsucmptnf	⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Proof of Theorem rdgsucmptnf

Step	Hyp	Ref	Expression
1		rdgsucmptf.4	. . 3 ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
2	1	fveq1i 6907	. 2 ⊢ (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵)
3		rdgdmlim 8457	. . . . 5 ⊢ Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
4		limsuc 7870	. . . . 5 ⊢ (Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↔ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)))
5	3, 4	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↔ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴))
6		rdgsucg 8463	. . . . . . 7 ⊢ (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)))
7	1	fveq1i 6907	. . . . . . . 8 ⊢ (𝐹‘𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)
8	7	fveq2i 6909	. . . . . . 7 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))
9	6, 8	eqtr4di 2795	. . . . . 6 ⊢ (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)))
10		nfmpt1 5250	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶)
11		rdgsucmptf.1	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
12	10, 11	nfrdg 8454	. . . . . . . . 9 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
13	1, 12	nfcxfr 2903	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
14		rdgsucmptf.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
15	13, 14	nffv 6916	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝐵)
16		rdgsucmptf.3	. . . . . . 7 ⊢ Ⅎ𝑥𝐷
17		rdgsucmptf.5	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)
18		eqid 2737	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
19	15, 16, 17, 18	fvmptnf 7038	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ∅)
20	9, 19	sylan9eqr 2799	. . . . 5 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
21	20	ex 412	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
22	5, 21	biimtrrid 243	. . 3 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
23		ndmfv 6941	. . 3 ⊢ (¬ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
24	22, 23	pm2.61d1 180	. 2 ⊢ (¬ 𝐷 ∈ V → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
25	2, 24	eqtrid 2789	1 ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2890  Vcvv 3480  ∅c0 4333   ↦ cmpt 5225  dom cdm 5685  Lim wlim 6385  suc csuc 6386  ‘cfv 6561  reccrdg 8449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450

This theorem is referenced by:  ttrclselem1  9765