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Theorem rdgprc0 33163
 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 6212 . . . 4 ∅ ∈ On
2 rdgval 8041 . . . 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 5822 . . . 4 (rec(𝐹, 𝐼) ↾ ∅) = ∅
54fveq2i 6648 . . 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 2821 . 2 (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘∅)
7 eqeq1 2802 . . . . . . 7 (𝑔 = ∅ → (𝑔 = ∅ ↔ ∅ = ∅))
8 dmeq 5736 . . . . . . . . 9 (𝑔 = ∅ → dom 𝑔 = dom ∅)
9 limeq 6171 . . . . . . . . 9 (dom 𝑔 = dom ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
108, 9syl 17 . . . . . . . 8 (𝑔 = ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
11 rneq 5770 . . . . . . . . 9 (𝑔 = ∅ → ran 𝑔 = ran ∅)
1211unieqd 4814 . . . . . . . 8 (𝑔 = ∅ → ran 𝑔 = ran ∅)
13 id 22 . . . . . . . . . 10 (𝑔 = ∅ → 𝑔 = ∅)
148unieqd 4814 . . . . . . . . . 10 (𝑔 = ∅ → dom 𝑔 = dom ∅)
1513, 14fveq12d 6652 . . . . . . . . 9 (𝑔 = ∅ → (𝑔 dom 𝑔) = (∅‘ dom ∅))
1615fveq2d 6649 . . . . . . . 8 (𝑔 = ∅ → (𝐹‘(𝑔 dom 𝑔)) = (𝐹‘(∅‘ dom ∅)))
1710, 12, 16ifbieq12d 4452 . . . . . . 7 (𝑔 = ∅ → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅))))
187, 17ifbieq2d 4450 . . . . . 6 (𝑔 = ∅ → if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) = if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))))
1918eleq1d 2874 . . . . 5 (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 2798 . . . . . 6 (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 6061 . . . . 5 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 3631 . . . 4 (∅ ∈ 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 2798 . . . . . . 7 ∅ = ∅
2423iftruei 4432 . . . . . 6 if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) = 𝐼
2524eleq1i 2880 . . . . 5 (if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V ↔ 𝐼 ∈ V)
2625biimpi 219 . . . 4 (if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V → 𝐼 ∈ V)
2722, 26simplbiim 508 . . 3 (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) → 𝐼 ∈ V)
28 ndmfv 6675 . . 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)))))‘∅) = ∅)
2927, 28nsyl5 162 . 2 𝐼 ∈ V → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘∅) = ∅)
306, 29syl5eq 2845 1 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  ifcif 4425  ∪ cuni 4800   ↦ cmpt 5110  dom cdm 5519  ran crn 5520   ↾ cres 5521  Oncon0 6159  Lim wlim 6160  ‘cfv 6324  reccrdg 8030 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-wrecs 7932  df-recs 7993  df-rdg 8031 This theorem is referenced by:  rdgprc  33164
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