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Theorem rdgprc0 30943
 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 5695 . . . 4 ∅ ∈ On
2 rdgval 7403 . . . 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 5321 . . . 4 (rec(𝐹, 𝐼) ↾ ∅) = ∅
54fveq2i 6106 . . 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 2632 . 2 (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘∅)
7 eqeq1 2614 . . . . . . . 8 (𝑔 = ∅ → (𝑔 = ∅ ↔ ∅ = ∅))
8 dmeq 5246 . . . . . . . . . 10 (𝑔 = ∅ → dom 𝑔 = dom ∅)
9 limeq 5652 . . . . . . . . . 10 (dom 𝑔 = dom ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
108, 9syl 17 . . . . . . . . 9 (𝑔 = ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
11 rneq 5272 . . . . . . . . . 10 (𝑔 = ∅ → ran 𝑔 = ran ∅)
1211unieqd 4382 . . . . . . . . 9 (𝑔 = ∅ → ran 𝑔 = ran ∅)
13 id 22 . . . . . . . . . . 11 (𝑔 = ∅ → 𝑔 = ∅)
148unieqd 4382 . . . . . . . . . . 11 (𝑔 = ∅ → dom 𝑔 = dom ∅)
1513, 14fveq12d 6109 . . . . . . . . . 10 (𝑔 = ∅ → (𝑔 dom 𝑔) = (∅‘ dom ∅))
1615fveq2d 6107 . . . . . . . . 9 (𝑔 = ∅ → (𝐹‘(𝑔 dom 𝑔)) = (𝐹‘(∅‘ dom ∅)))
1710, 12, 16ifbieq12d 4063 . . . . . . . 8 (𝑔 = ∅ → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅))))
187, 17ifbieq2d 4061 . . . . . . 7 (𝑔 = ∅ → if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) = if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))))
1918eleq1d 2672 . . . . . 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 2610 . . . . . . 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 5547 . . . . . 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 3333 . . . . 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 2610 . . . . . . . . 9 ∅ = ∅
2423iftruei 4043 . . . . . . . 8 if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) = 𝐼
2524eleq1i 2679 . . . . . . 7 (if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V ↔ 𝐼 ∈ V)
2625biimpi 205 . . . . . 6 (if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V → 𝐼 ∈ V)
2726adantl 481 . . . . 5 ((∅ ∈ V ∧ if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V) → 𝐼 ∈ V)
2822, 27sylbi 206 . . . 4 (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) → 𝐼 ∈ V)
2928con3i 149 . . 3 𝐼 ∈ V → ¬ ∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
30 ndmfv 6128 . . 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 2656 1 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ifcif 4036  ∪ cuni 4372   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   ↾ cres 5040  Oncon0 5640  Lim wlim 5641  ‘cfv 5804  reccrdg 7392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355  df-rdg 7393 This theorem is referenced by:  rdgprc  30944
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