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Theorem rdgsucmptnf 7677
 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 7676 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 6333 . 2 (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵)
3 rdgdmlim 7665 . . . . 5 Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
4 limsuc 7195 . . . . 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 7671 . . . . . . 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 6333 . . . . . . . 8 (𝐹𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)
87fveq2i 6335 . . . . . . 7 ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))
96, 8syl6eqr 2822 . . . . . 6 (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)))
10 nfmpt1 4879 . . . . . . . . . 10 𝑥(𝑥 ∈ V ↦ 𝐶)
11 rdgsucmptf.1 . . . . . . . . . 10 𝑥𝐴
1210, 11nfrdg 7662 . . . . . . . . 9 𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
131, 12nfcxfr 2910 . . . . . . . 8 𝑥𝐹
14 rdgsucmptf.2 . . . . . . . 8 𝑥𝐵
1513, 14nffv 6339 . . . . . . 7 𝑥(𝐹𝐵)
16 rdgsucmptf.3 . . . . . . 7 𝑥𝐷
17 rdgsucmptf.5 . . . . . . 7 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
18 eqid 2770 . . . . . . 7 (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
1915, 16, 17, 18fvmptnf 6444 . . . . . 6 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ∅)
209, 19sylan9eqr 2826 . . . . 5 ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
2120ex 397 . . . 4 𝐷 ∈ V → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
225, 21syl5bir 233 . . 3 𝐷 ∈ V → (suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
23 ndmfv 6359 . . 3 (¬ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
2422, 23pm2.61d1 172 . 2 𝐷 ∈ V → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
252, 24syl5eq 2816 1 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1630   ∈ wcel 2144  Ⅎwnfc 2899  Vcvv 3349  ∅c0 4061   ↦ cmpt 4861  dom cdm 5249  Lim wlim 5867  suc csuc 5868  ‘cfv 6031  reccrdg 7657 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-wrecs 7558  df-recs 7620  df-rdg 7658 This theorem is referenced by: (None)
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