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Theorem rdgprc0 32649
 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 6126 . . . 4 ∅ ∈ On
2 rdgval 7915 . . . 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 5745 . . . 4 (rec(𝐹, 𝐼) ↾ ∅) = ∅
54fveq2i 6548 . . 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 2801 . . . . . . . 8 (𝑔 = ∅ → (𝑔 = ∅ ↔ ∅ = ∅))
8 dmeq 5665 . . . . . . . . . 10 (𝑔 = ∅ → dom 𝑔 = dom ∅)
9 limeq 6085 . . . . . . . . . 10 (dom 𝑔 = dom ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
108, 9syl 17 . . . . . . . . 9 (𝑔 = ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
11 rneq 5695 . . . . . . . . . 10 (𝑔 = ∅ → ran 𝑔 = ran ∅)
1211unieqd 4761 . . . . . . . . 9 (𝑔 = ∅ → ran 𝑔 = ran ∅)
13 id 22 . . . . . . . . . . 11 (𝑔 = ∅ → 𝑔 = ∅)
148unieqd 4761 . . . . . . . . . . 11 (𝑔 = ∅ → dom 𝑔 = dom ∅)
1513, 14fveq12d 6552 . . . . . . . . . 10 (𝑔 = ∅ → (𝑔 dom 𝑔) = (∅‘ dom ∅))
1615fveq2d 6549 . . . . . . . . 9 (𝑔 = ∅ → (𝐹‘(𝑔 dom 𝑔)) = (𝐹‘(∅‘ dom ∅)))
1710, 12, 16ifbieq12d 4414 . . . . . . . 8 (𝑔 = ∅ → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅))))
187, 17ifbieq2d 4412 . . . . . . 7 (𝑔 = ∅ → if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) = if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))))
1918eleq1d 2869 . . . . . 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 2797 . . . . . . 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 5976 . . . . . 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 3624 . . . . 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 2797 . . . . . . . . 9 ∅ = ∅
2423iftruei 4394 . . . . . . . 8 if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) = 𝐼
2524eleq1i 2875 . . . . . . 7 (if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V ↔ 𝐼 ∈ V)
2625biimpi 217 . . . . . 6 (if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V → 𝐼 ∈ V)
2726adantl 482 . . . . 5 ((∅ ∈ V ∧ if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V) → 𝐼 ∈ V)
2822, 27sylbi 218 . . . 4 (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) → 𝐼 ∈ V)
2928con3i 157 . . 3 𝐼 ∈ V → ¬ ∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
30 ndmfv 6575 . . 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 2845 1 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1525   ∈ wcel 2083  Vcvv 3440  ∅c0 4217  ifcif 4387  ∪ cuni 4751   ↦ cmpt 5047  dom cdm 5450  ran crn 5451   ↾ cres 5452  Oncon0 6073  Lim wlim 6074  ‘cfv 6232  reccrdg 7904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-wrecs 7805  df-recs 7867  df-rdg 7905 This theorem is referenced by:  rdgprc  32650
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