Theorem frsucmptn 8368

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 8367 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 6828	. 2 ⊢ (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵)
3		frfnom 8364	. . . . . 6 ⊢ (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) Fn ω
4		fndm 6588	. . . . . 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 2831	. . . 4 ⊢ (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) ↔ suc 𝐵 ∈ ω)
7		peano2b 7823	. . . . 5 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
8		frsuc 8366	. . . . . . . 8 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)))
9	1	fveq1i 6828	. . . . . . . . 9 ⊢ (𝐹‘𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)
10	9	fveq2i 6830	. . . . . . . 8 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))
11	8, 10	eqtr4di 2792	. . . . . . 7 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)))
12		nfmpt1 5171	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶)
13		frsucmpt.1	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐴
14	12, 13	nfrdg 8343	. . . . . . . . . . 11 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
15		nfcv 2901	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ω
16	14, 15	nfres 5933	. . . . . . . . . 10 ⊢ Ⅎ𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
17	1, 16	nfcxfr 2899	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
18		frsucmpt.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐵
19	17, 18	nffv 6837	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝐵)
20		frsucmpt.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐷
21		frsucmpt.5	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)
22		eqid 2739	. . . . . . . 8 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
23	19, 20, 21, 22	fvmptnf 6958	. . . . . . 7 ⊢ (¬ 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ∅)
24	11, 23	sylan9eqr 2796	. . . . . 6 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
25	24	ex 413	. . . . 5 ⊢ (¬ 𝐷 ∈ V → (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
26	7, 25	biimtrrid 244	. . . 4 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
27	6, 26	biimtrid 243	. . 3 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
28		ndmfv 6859	. . 3 ⊢ (¬ suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
29	27, 28	pm2.61d1 181	. 2 ⊢ (¬ 𝐷 ∈ V → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
30	2, 29	eqtrid 2786	1 ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1547   ∈ wcel 2119  Ⅎwnfc 2886  Vcvv 3431  ∅c0 4261   ↦ cmpt 5153  dom cdm 5618   ↾ cres 5620  suc csuc 6312   Fn wfn 6480  ‘cfv 6485  ωcom 7806  reccrdg 8338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339

This theorem is referenced by: (None)