Theorem frsucmpt 8072

Description: The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation). (Contributed by NM, 14-Sep-2003.) (Revised by Scott Fenton, 2-Nov-2011.)

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
frsucmpt	⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Proof of Theorem frsucmpt

Step	Hyp	Ref	Expression
1		frsuc 8071	. . 3 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)))
2		frsucmpt.4	. . . 4 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
3	2	fveq1i 6670	. . 3 ⊢ (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵)
4	2	fveq1i 6670	. . . 4 ⊢ (𝐹‘𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)
5	4	fveq2i 6672	. . 3 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))
6	1, 3, 5	3eqtr4g 2881	. 2 ⊢ (𝐵 ∈ ω → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)))
7		fvex 6682	. . 3 ⊢ (𝐹‘𝐵) ∈ V
8		nfmpt1 5163	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶)
9		frsucmpt.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
10	8, 9	nfrdg 8049	. . . . . . 7 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
11		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑥ω
12	10, 11	nfres 5854	. . . . . 6 ⊢ Ⅎ𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
13	2, 12	nfcxfr 2975	. . . . 5 ⊢ Ⅎ𝑥𝐹
14		frsucmpt.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
15	13, 14	nffv 6679	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐵)
16		frsucmpt.3	. . . 4 ⊢ Ⅎ𝑥𝐷
17		frsucmpt.5	. . . 4 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)
18		eqid 2821	. . . 4 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
19	15, 16, 17, 18	fvmptf 6788	. . 3 ⊢ (((𝐹‘𝐵) ∈ V ∧ 𝐷 ∈ 𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷)
20	7, 19	mpan 688	. 2 ⊢ (𝐷 ∈ 𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷)
21	6, 20	sylan9eq 2876	1 ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Ⅎwnfc 2961  Vcvv 3494   ↦ cmpt 5145   ↾ cres 5556  suc csuc 6192  ‘cfv 6354  ωcom 7579  reccrdg 8044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045

This theorem is referenced by:  frsucmpt2w  8074  frsucmpt2  8075  dffi3  8894  axdclem  9940  trpredlem1  33066  trpredtr  33069  trpredmintr  33070  trpred0  33075  trpredrec  33077