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Theorem rdgprc0 32074
Description: The value of the recursive definition generator at when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rdgprc0 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)

Proof of Theorem rdgprc0
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 0elon 5961 . . . 4 ∅ ∈ On
2 rdgval 7720 . . . 4 (∅ ∈ On → (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(rec(𝐹, 𝐼) ↾ ∅)))
31, 2ax-mp 5 . . 3 (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(rec(𝐹, 𝐼) ↾ ∅))
4 res0 5569 . . . 4 (rec(𝐹, 𝐼) ↾ ∅) = ∅
54fveq2i 6378 . . 3 ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(rec(𝐹, 𝐼) ↾ ∅)) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘∅)
63, 5eqtri 2787 . 2 (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘∅)
7 eqeq1 2769 . . . . . . . 8 (𝑔 = ∅ → (𝑔 = ∅ ↔ ∅ = ∅))
8 dmeq 5492 . . . . . . . . . 10 (𝑔 = ∅ → dom 𝑔 = dom ∅)
9 limeq 5920 . . . . . . . . . 10 (dom 𝑔 = dom ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
108, 9syl 17 . . . . . . . . 9 (𝑔 = ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
11 rneq 5519 . . . . . . . . . 10 (𝑔 = ∅ → ran 𝑔 = ran ∅)
1211unieqd 4604 . . . . . . . . 9 (𝑔 = ∅ → ran 𝑔 = ran ∅)
13 id 22 . . . . . . . . . . 11 (𝑔 = ∅ → 𝑔 = ∅)
148unieqd 4604 . . . . . . . . . . 11 (𝑔 = ∅ → dom 𝑔 = dom ∅)
1513, 14fveq12d 6382 . . . . . . . . . 10 (𝑔 = ∅ → (𝑔 dom 𝑔) = (∅‘ dom ∅))
1615fveq2d 6379 . . . . . . . . 9 (𝑔 = ∅ → (𝐹‘(𝑔 dom 𝑔)) = (𝐹‘(∅‘ dom ∅)))
1710, 12, 16ifbieq12d 4270 . . . . . . . 8 (𝑔 = ∅ → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅))))
187, 17ifbieq2d 4268 . . . . . . 7 (𝑔 = ∅ → if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) = if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))))
1918eleq1d 2829 . . . . . 6 (𝑔 = ∅ → (if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) ∈ V ↔ if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V))
20 eqid 2765 . . . . . . 7 (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
2120dmmpt 5816 . . . . . 6 dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) = {𝑔 ∈ V ∣ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) ∈ V}
2219, 21elrab2 3523 . . . . 5 (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) ↔ (∅ ∈ V ∧ if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V))
23 eqid 2765 . . . . . . . . 9 ∅ = ∅
2423iftruei 4250 . . . . . . . 8 if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) = 𝐼
2524eleq1i 2835 . . . . . . 7 (if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V ↔ 𝐼 ∈ V)
2625biimpi 207 . . . . . 6 (if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V → 𝐼 ∈ V)
2726adantl 473 . . . . 5 ((∅ ∈ V ∧ if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V) → 𝐼 ∈ V)
2822, 27sylbi 208 . . . 4 (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) → 𝐼 ∈ V)
2928con3i 151 . . 3 𝐼 ∈ V → ¬ ∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
30 ndmfv 6405 . . 3 (¬ ∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘∅) = ∅)
3129, 30syl 17 . 2 𝐼 ∈ V → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘∅) = ∅)
326, 31syl5eq 2811 1 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  c0 4079  ifcif 4243   cuni 4594  cmpt 4888  dom cdm 5277  ran crn 5278  cres 5279  Oncon0 5908  Lim wlim 5909  cfv 6068  reccrdg 7709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-wrecs 7610  df-recs 7672  df-rdg 7710
This theorem is referenced by:  rdgprc  32075
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