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Theorem rdgsucmptf 7411
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 7407 . . 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 6104 . . 3 (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵)
42fveq1i 6104 . . . 4 (𝐹𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)
54fveq2i 6106 . . 3 ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))
61, 3, 53eqtr4g 2669 . 2 (𝐵 ∈ On → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)))
7 fvex 6113 . . 3 (𝐹𝐵) ∈ V
8 nfmpt1 4675 . . . . . . 7 𝑥(𝑥 ∈ V ↦ 𝐶)
9 rdgsucmptf.1 . . . . . . 7 𝑥𝐴
108, 9nfrdg 7397 . . . . . 6 𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
112, 10nfcxfr 2749 . . . . 5 𝑥𝐹
12 rdgsucmptf.2 . . . . 5 𝑥𝐵
1311, 12nffv 6110 . . . 4 𝑥(𝐹𝐵)
14 rdgsucmptf.3 . . . 4 𝑥𝐷
15 rdgsucmptf.5 . . . 4 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
16 eqid 2610 . . . 4 (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
1713, 14, 15, 16fvmptf 6209 . . 3 (((𝐹𝐵) ∈ V ∧ 𝐷𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = 𝐷)
187, 17mpan 702 . 2 (𝐷𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = 𝐷)
196, 18sylan9eq 2664 1 ((𝐵 ∈ On ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wnfc 2738  Vcvv 3173  cmpt 4643  Oncon0 5640  suc csuc 5642  cfv 5804  reccrdg 7392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355  df-rdg 7393
This theorem is referenced by:  rdgsucmpt2  7413  rdgsucmpt  7414
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