Theorem frsucmptn 7579

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 7578 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 6230	. 2 ⊢ (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵)
3		frfnom 7575	. . . . . 6 ⊢ (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) Fn ω
4		fndm 6028	. . . . . 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 2722	. . . 4 ⊢ (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) ↔ suc 𝐵 ∈ ω)
7		peano2b 7123	. . . . 5 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
8		frsuc 7577	. . . . . . . 8 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)))
9	1	fveq1i 6230	. . . . . . . . 9 ⊢ (𝐹‘𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)
10	9	fveq2i 6232	. . . . . . . 8 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))
11	8, 10	syl6eqr 2703	. . . . . . 7 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)))
12		nfmpt1 4780	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶)
13		frsucmpt.1	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐴
14	12, 13	nfrdg 7555	. . . . . . . . . . 11 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
15		nfcv 2793	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ω
16	14, 15	nfres 5430	. . . . . . . . . 10 ⊢ Ⅎ𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
17	1, 16	nfcxfr 2791	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
18		frsucmpt.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐵
19	17, 18	nffv 6236	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝐵)
20		frsucmpt.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝐷
21		frsucmpt.5	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)
22		eqid 2651	. . . . . . . 8 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
23	19, 20, 21, 22	fvmptnf 6341	. . . . . . 7 ⊢ (¬ 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ∅)
24	11, 23	sylan9eqr 2707	. . . . . 6 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
25	24	ex 449	. . . . 5 ⊢ (¬ 𝐷 ∈ V → (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
26	7, 25	syl5bir 233	. . . 4 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
27	6, 26	syl5bi 232	. . 3 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅))
28		ndmfv 6256	. . 3 ⊢ (¬ suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
29	27, 28	pm2.61d1 171	. 2 ⊢ (¬ 𝐷 ∈ V → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)
30	2, 29	syl5eq 2697	1 ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523   ∈ wcel 2030  Ⅎwnfc 2780  Vcvv 3231  ∅c0 3948   ↦ cmpt 4762  dom cdm 5143   ↾ cres 5145  suc csuc 5763   Fn wfn 5921  ‘cfv 5926  ωcom 7107  reccrdg 7550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551

This theorem is referenced by:  trpredlem1  31851