Theorem rdgsucmptnf 8431

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 8430 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 6891	. 2 ⊢ (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵)
3		rdgdmlim 8419	. . . . 5 ⊢ Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
4		limsuc 7840	. . . . 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 8425	. . . . . . 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 6891	. . . . . . . 8 ⊢ (𝐹‘𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)
8	7	fveq2i 6893	. . . . . . 7 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))
9	6, 8	eqtr4di 2788	. . . . . 6 ⊢ (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)))
10		nfmpt1 5255	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶)
11		rdgsucmptf.1	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
12	10, 11	nfrdg 8416	. . . . . . . . 9 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
13	1, 12	nfcxfr 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
14		rdgsucmptf.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
15	13, 14	nffv 6900	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝐵)
16		rdgsucmptf.3	. . . . . . 7 ⊢ Ⅎ𝑥𝐷
17		rdgsucmptf.5	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)
18		eqid 2730	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
19	15, 16, 17, 18	fvmptnf 7019	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ∅)
20	9, 19	sylan9eqr 2792	. . . . 5 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
21	20	ex 411	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
22	5, 21	biimtrrid 242	. . 3 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
23		ndmfv 6925	. . 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 2782	1 ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2104  Ⅎwnfc 2881  Vcvv 3472  ∅c0 4321   ↦ cmpt 5230  dom cdm 5675  Lim wlim 6364  suc csuc 6365  ‘cfv 6542  reccrdg 8411

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412

This theorem is referenced by:  ttrclselem1  9722