Theorem rdgprc0 35766

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.

Step	Hyp	Ref	Expression
1		0elon 6366	. . . 4 ⊢ ∅ ∈ On
2		rdgval 8349	. . . 4 ⊢ (∅ ∈ On → (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(rec(𝐹, 𝐼) ↾ ∅)))
3	1, 2	ax-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 5938	. . . 4 ⊢ (rec(𝐹, 𝐼) ↾ ∅) = ∅
5	4	fveq2i 6829	. . 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)))))‘∅)
6	3, 5	eqtri 2752	. 2 ⊢ (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘∅)
7		eqeq1 2733	. . . . . . 7 ⊢ (𝑔 = ∅ → (𝑔 = ∅ ↔ ∅ = ∅))
8		dmeq 5850	. . . . . . . . 9 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
9		limeq 6323	. . . . . . . . 9 ⊢ (dom 𝑔 = dom ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝑔 = ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
11		rneq 5882	. . . . . . . . 9 ⊢ (𝑔 = ∅ → ran 𝑔 = ran ∅)
12	11	unieqd 4874	. . . . . . . 8 ⊢ (𝑔 = ∅ → ∪ ran 𝑔 = ∪ ran ∅)
13		id 22	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → 𝑔 = ∅)
14	8	unieqd 4874	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → ∪ dom 𝑔 = ∪ dom ∅)
15	13, 14	fveq12d 6833	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔‘∪ dom 𝑔) = (∅‘∪ dom ∅))
16	15	fveq2d 6830	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐹‘(∅‘∪ dom ∅)))
17	10, 12, 16	ifbieq12d 4507	. . . . . . 7 ⊢ (𝑔 = ∅ → if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) = if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅))))
18	7, 17	ifbieq2d 4505	. . . . . 6 ⊢ (𝑔 = ∅ → if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) = if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))))
19	18	eleq1d 2813	. . . . 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 2729	. . . . . 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)))))
21	20	dmmpt 6193	. . . . 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}
22	19, 21	elrab2 3653	. . . 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 2729	. . . . . . 7 ⊢ ∅ = ∅
24	23	iftruei 4485	. . . . . 6 ⊢ if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) = 𝐼
25	24	eleq1i 2819	. . . . 5 ⊢ (if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V ↔ 𝐼 ∈ V)
26	25	biimpi 216	. . . 4 ⊢ (if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V → 𝐼 ∈ V)
27	22, 26	simplbiim 504	. . 3 ⊢ (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) → 𝐼 ∈ V)
28		ndmfv 6859	. . 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)))))‘∅) = ∅)
29	27, 28	nsyl5 159	. 2 ⊢ (¬ 𝐼 ∈ V → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘∅) = ∅)
30	6, 29	eqtrid 2776	1 ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ∅c0 4286  ifcif 4478  ∪ cuni 4861   ↦ cmpt 5176  dom cdm 5623  ran crn 5624   ↾ cres 5625  Oncon0 6311  Lim wlim 6312  ‘cfv 6486  reccrdg 8338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339

This theorem is referenced by:  rdgprc  35767