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Theorem rdgsucmptf 8429
Description: The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
rdgsucmptf.1 𝑥𝐴
rdgsucmptf.2 𝑥𝐵
rdgsucmptf.3 𝑥𝐷
rdgsucmptf.4 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
rdgsucmptf.5 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
rdgsucmptf ((𝐵 ∈ On ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Proof of Theorem rdgsucmptf
StepHypRef Expression
1 rdgsuc 8425 . . 3 (𝐵 ∈ On → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)))
2 rdgsucmptf.4 . . . 4 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
32fveq1i 6886 . . 3 (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵)
42fveq1i 6886 . . . 4 (𝐹𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)
54fveq2i 6888 . . 3 ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))
61, 3, 53eqtr4g 2791 . 2 (𝐵 ∈ On → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)))
7 fvex 6898 . . 3 (𝐹𝐵) ∈ V
8 nfmpt1 5249 . . . . . . 7 𝑥(𝑥 ∈ V ↦ 𝐶)
9 rdgsucmptf.1 . . . . . . 7 𝑥𝐴
108, 9nfrdg 8415 . . . . . 6 𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
112, 10nfcxfr 2895 . . . . 5 𝑥𝐹
12 rdgsucmptf.2 . . . . 5 𝑥𝐵
1311, 12nffv 6895 . . . 4 𝑥(𝐹𝐵)
14 rdgsucmptf.3 . . . 4 𝑥𝐷
15 rdgsucmptf.5 . . . 4 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
16 eqid 2726 . . . 4 (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
1713, 14, 15, 16fvmptf 7013 . . 3 (((𝐹𝐵) ∈ V ∧ 𝐷𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = 𝐷)
187, 17mpan 687 . 2 (𝐷𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = 𝐷)
196, 18sylan9eq 2786 1 ((𝐵 ∈ On ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wnfc 2877  Vcvv 3468  cmpt 5224  Oncon0 6358  suc csuc 6360  cfv 6537  reccrdg 8410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411
This theorem is referenced by:  rdgsucmpt2  8431  rdgsucmpt  8432  ttrclselem1  9722  ttrclselem2  9723  rdgssun  36766  exrecfnlem  36767
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