Theorem frsucmpt 8494

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 8493	. . 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 6921	. . 3 ⊢ (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵)
4	2	fveq1i 6921	. . . 4 ⊢ (𝐹‘𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)
5	4	fveq2i 6923	. . 3 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))
6	1, 3, 5	3eqtr4g 2805	. 2 ⊢ (𝐵 ∈ ω → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)))
7		fvex 6933	. . 3 ⊢ (𝐹‘𝐵) ∈ V
8		nfmpt1 5274	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶)
9		frsucmpt.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
10	8, 9	nfrdg 8470	. . . . . . 7 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
11		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥ω
12	10, 11	nfres 6011	. . . . . 6 ⊢ Ⅎ𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
13	2, 12	nfcxfr 2906	. . . . 5 ⊢ Ⅎ𝑥𝐹
14		frsucmpt.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
15	13, 14	nffv 6930	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐵)
16		frsucmpt.3	. . . 4 ⊢ Ⅎ𝑥𝐷
17		frsucmpt.5	. . . 4 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)
18		eqid 2740	. . . 4 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
19	15, 16, 17, 18	fvmptf 7050	. . 3 ⊢ (((𝐹‘𝐵) ∈ V ∧ 𝐷 ∈ 𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷)
20	7, 19	mpan 689	. 2 ⊢ (𝐷 ∈ 𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷)
21	6, 20	sylan9eq 2800	1 ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Ⅎwnfc 2893  Vcvv 3488   ↦ cmpt 5249   ↾ cres 5702  suc csuc 6397  ‘cfv 6573  ωcom 7903  reccrdg 8465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466

This theorem is referenced by:  frsucmpt2  8496  dffi3  9500  axdclem  10588