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Theorem rdgsucmptnf 8469
Description: The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class 𝐷 is a proper class). This is a technical lemma that can be used together with rdgsucmptf 8468 to help eliminate redundant sethood antecedents. (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
rdgsucmptnf 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Proof of Theorem rdgsucmptnf
StepHypRef Expression
1 rdgsucmptf.4 . . 3 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
21fveq1i 6907 . 2 (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵)
3 rdgdmlim 8457 . . . . 5 Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
4 limsuc 7870 . . . . 5 (Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↔ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)))
53, 4ax-mp 5 . . . 4 (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↔ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴))
6 rdgsucg 8463 . . . . . . 7 (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)))
71fveq1i 6907 . . . . . . . 8 (𝐹𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)
87fveq2i 6909 . . . . . . 7 ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))
96, 8eqtr4di 2795 . . . . . 6 (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)))
10 nfmpt1 5250 . . . . . . . . . 10 𝑥(𝑥 ∈ V ↦ 𝐶)
11 rdgsucmptf.1 . . . . . . . . . 10 𝑥𝐴
1210, 11nfrdg 8454 . . . . . . . . 9 𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
131, 12nfcxfr 2903 . . . . . . . 8 𝑥𝐹
14 rdgsucmptf.2 . . . . . . . 8 𝑥𝐵
1513, 14nffv 6916 . . . . . . 7 𝑥(𝐹𝐵)
16 rdgsucmptf.3 . . . . . . 7 𝑥𝐷
17 rdgsucmptf.5 . . . . . . 7 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
18 eqid 2737 . . . . . . 7 (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
1915, 16, 17, 18fvmptnf 7038 . . . . . 6 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ∅)
209, 19sylan9eqr 2799 . . . . 5 ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
2120ex 412 . . . 4 𝐷 ∈ V → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
225, 21biimtrrid 243 . . 3 𝐷 ∈ V → (suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
23 ndmfv 6941 . . 3 (¬ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
2422, 23pm2.61d1 180 . 2 𝐷 ∈ V → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
252, 24eqtrid 2789 1 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1540  wcel 2108  wnfc 2890  Vcvv 3480  c0 4333  cmpt 5225  dom cdm 5685  Lim wlim 6385  suc csuc 6386  cfv 6561  reccrdg 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  ttrclselem1  9765
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