Theorem frsucmptn 8477

Description: The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class 𝐷 is a proper class). This is a technical lemma that can be used together with frsucmpt 8476 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypotheses

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

Assertion

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

Proof of Theorem frsucmptn

Step	Hyp	Ref	Expression
1		frsucmpt.4	. . 3 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
2	1	fveq1i 6906	. 2 ⊢ (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵)
3		frfnom 8473	. . . . . 6 ⊢ (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) Fn ω
4		fndm 6667	. . . . . 6 ⊢ ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) Fn ω → dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) = ω)
5	3, 4	ax-mp 5	. . . . 5 ⊢ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) = ω
6	5	eleq2i 2821	. . . 4 ⊢ (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) ↔ suc 𝐵 ∈ ω)
7		peano2b 7901	. . . . 5 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
8		frsuc 8475	. . . . . . . 8 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)))
9	1	fveq1i 6906	. . . . . . . . 9 ⊢ (𝐹‘𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)
10	9	fveq2i 6908	. . . . . . . 8 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))
11	8, 10	eqtr4di 2787	. . . . . . 7 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)))
12		nfmpt1 5264	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶)
13		frsucmpt.1	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐴
14	12, 13	nfrdg 8452	. . . . . . . . . . 11 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
15		nfcv 2895	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ω
16	14, 15	nfres 5996	. . . . . . . . . 10 ⊢ Ⅎ𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
17	1, 16	nfcxfr 2893	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
18		frsucmpt.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐵
19	17, 18	nffv 6915	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝐵)
20		frsucmpt.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐷
21		frsucmpt.5	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)
22		eqid 2729	. . . . . . . 8 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
23	19, 20, 21, 22	fvmptnf 7036	. . . . . . 7 ⊢ (¬ 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ∅)
24	11, 23	sylan9eqr 2791	. . . . . 6 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
25	24	ex 411	. . . . 5 ⊢ (¬ 𝐷 ∈ V → (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
26	7, 25	biimtrrid 242	. . . 4 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
27	6, 26	biimtrid 241	. . 3 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
28		ndmfv 6940	. . 3 ⊢ (¬ suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
29	27, 28	pm2.61d1 180	. 2 ⊢ (¬ 𝐷 ∈ V → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
30	2, 29	eqtrid 2781	1 ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1534   ∈ wcel 2100  Ⅎwnfc 2879  Vcvv 3472  ∅c0 4335   ↦ cmpt 5239  dom cdm 5686   ↾ cres 5688  suc csuc 6382   Fn wfn 6553  ‘cfv 6558  ωcom 7884  reccrdg 8447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5307  ax-nul 5314  ax-pr 5437  ax-un 7751

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3789  df-csb 3905  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-pss 3979  df-nul 4336  df-if 4537  df-pw 4612  df-sn 4637  df-pr 4639  df-op 4643  df-uni 4919  df-iun 5008  df-br 5157  df-opab 5219  df-mpt 5240  df-tr 5274  df-id 5584  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5641  df-we 5643  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-ima 5699  df-pred 6317  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7433  df-om 7885  df-2nd 8012  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448

This theorem is referenced by: (None)