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Theorem rdgsucmptnf 8052
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 8051 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 6650 . 2 (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵)
3 rdgdmlim 8040 . . . . 5 Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
4 limsuc 7548 . . . . 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 8046 . . . . . . 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 6650 . . . . . . . 8 (𝐹𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)
87fveq2i 6652 . . . . . . 7 ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))
96, 8eqtr4di 2854 . . . . . 6 (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)))
10 nfmpt1 5131 . . . . . . . . . 10 𝑥(𝑥 ∈ V ↦ 𝐶)
11 rdgsucmptf.1 . . . . . . . . . 10 𝑥𝐴
1210, 11nfrdg 8037 . . . . . . . . 9 𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
131, 12nfcxfr 2956 . . . . . . . 8 𝑥𝐹
14 rdgsucmptf.2 . . . . . . . 8 𝑥𝐵
1513, 14nffv 6659 . . . . . . 7 𝑥(𝐹𝐵)
16 rdgsucmptf.3 . . . . . . 7 𝑥𝐷
17 rdgsucmptf.5 . . . . . . 7 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
18 eqid 2801 . . . . . . 7 (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
1915, 16, 17, 18fvmptnf 6771 . . . . . 6 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ∅)
209, 19sylan9eqr 2858 . . . . 5 ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
2120ex 416 . . . 4 𝐷 ∈ V → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
225, 21syl5bir 246 . . 3 𝐷 ∈ V → (suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
23 ndmfv 6679 . . 3 (¬ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
2422, 23pm2.61d1 183 . 2 𝐷 ∈ V → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
252, 24syl5eq 2848 1 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  wcel 2112  wnfc 2939  Vcvv 3444  c0 4246  cmpt 5113  dom cdm 5523  Lim wlim 6164  suc csuc 6165  cfv 6328  reccrdg 8032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-wrecs 7934  df-recs 7995  df-rdg 8033
This theorem is referenced by: (None)
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