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Theorem frsucmpt 8409
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 𝑥𝐴
frsucmpt.2 𝑥𝐵
frsucmpt.3 𝑥𝐷
frsucmpt.4 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
frsucmpt.5 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
frsucmpt ((𝐵 ∈ ω ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Proof of Theorem frsucmpt
StepHypRef Expression
1 frsuc 8408 . . 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)
32fveq1i 6868 . . 3 (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵)
42fveq1i 6868 . . . 4 (𝐹𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)
54fveq2i 6870 . . 3 ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))
61, 3, 53eqtr4g 2822 . 2 (𝐵 ∈ ω → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)))
7 fvex 6880 . . 3 (𝐹𝐵) ∈ V
8 nfmpt1 5199 . . . . . . . 8 𝑥(𝑥 ∈ V ↦ 𝐶)
9 frsucmpt.1 . . . . . . . 8 𝑥𝐴
108, 9nfrdg 8385 . . . . . . 7 𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
11 nfcv 2924 . . . . . . 7 𝑥ω
1210, 11nfres 5967 . . . . . 6 𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
132, 12nfcxfr 2922 . . . . 5 𝑥𝐹
14 frsucmpt.2 . . . . 5 𝑥𝐵
1513, 14nffv 6877 . . . 4 𝑥(𝐹𝐵)
16 frsucmpt.3 . . . 4 𝑥𝐷
17 frsucmpt.5 . . . 4 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
18 eqid 2762 . . . 4 (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
1915, 16, 17, 18fvmptf 6997 . . 3 (((𝐹𝐵) ∈ V ∧ 𝐷𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = 𝐷)
207, 19mpan 700 . 2 (𝐷𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = 𝐷)
216, 20sylan9eq 2817 1 ((𝐵 ∈ ω ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1560  wcel 2142  wnfc 2909  Vcvv 3454  cmpt 5181  cres 5649  suc csuc 6348  cfv 6521  ωcom 7846  reccrdg 8380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381
This theorem is referenced by:  frsucmpt2  8411  dffi3  9377  axdclem  10476
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