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Theorem rdgprc0 34368
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 6371 . . . 4 ∅ ∈ On
2 rdgval 8366 . . . 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 5941 . . . 4 (rec(𝐹, 𝐼) ↾ ∅) = ∅
54fveq2i 6845 . . 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 2764 . 2 (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘∅)
7 eqeq1 2740 . . . . . . 7 (𝑔 = ∅ → (𝑔 = ∅ ↔ ∅ = ∅))
8 dmeq 5859 . . . . . . . . 9 (𝑔 = ∅ → dom 𝑔 = dom ∅)
9 limeq 6329 . . . . . . . . 9 (dom 𝑔 = dom ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
108, 9syl 17 . . . . . . . 8 (𝑔 = ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
11 rneq 5891 . . . . . . . . 9 (𝑔 = ∅ → ran 𝑔 = ran ∅)
1211unieqd 4879 . . . . . . . 8 (𝑔 = ∅ → ran 𝑔 = ran ∅)
13 id 22 . . . . . . . . . 10 (𝑔 = ∅ → 𝑔 = ∅)
148unieqd 4879 . . . . . . . . . 10 (𝑔 = ∅ → dom 𝑔 = dom ∅)
1513, 14fveq12d 6849 . . . . . . . . 9 (𝑔 = ∅ → (𝑔 dom 𝑔) = (∅‘ dom ∅))
1615fveq2d 6846 . . . . . . . 8 (𝑔 = ∅ → (𝐹‘(𝑔 dom 𝑔)) = (𝐹‘(∅‘ dom ∅)))
1710, 12, 16ifbieq12d 4514 . . . . . . 7 (𝑔 = ∅ → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅))))
187, 17ifbieq2d 4512 . . . . . 6 (𝑔 = ∅ → if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) = if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))))
1918eleq1d 2822 . . . . 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 2736 . . . . . 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 6192 . . . . 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 3648 . . . 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 2736 . . . . . . 7 ∅ = ∅
2423iftruei 4493 . . . . . 6 if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) = 𝐼
2524eleq1i 2828 . . . . 5 (if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V ↔ 𝐼 ∈ V)
2625biimpi 215 . . . 4 (if(∅ = ∅, 𝐼, if(Lim dom ∅, ran ∅, (𝐹‘(∅‘ dom ∅)))) ∈ V → 𝐼 ∈ V)
2722, 26simplbiim 505 . . 3 (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) → 𝐼 ∈ V)
28 ndmfv 6877 . . 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 159 . 2 𝐼 ∈ V → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘∅) = ∅)
306, 29eqtrid 2788 1 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1541  wcel 2106  Vcvv 3445  c0 4282  ifcif 4486   cuni 4865  cmpt 5188  dom cdm 5633  ran crn 5634  cres 5635  Oncon0 6317  Lim wlim 6318  cfv 6496  reccrdg 8355
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356
This theorem is referenced by:  rdgprc  34369
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