Theorem rdgprc0 33040

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 6246	. . . 4 ⊢ ∅ ∈ On
2		rdgval 8058	. . . 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 5859	. . . 4 ⊢ (rec(𝐹, 𝐼) ↾ ∅) = ∅
5	4	fveq2i 6675	. . 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 2846	. 2 ⊢ (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘∅)
7		eqeq1 2827	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝑔 = ∅ ↔ ∅ = ∅))
8		dmeq 5774	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
9		limeq 6205	. . . . . . . . . 10 ⊢ (dom 𝑔 = dom ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
11		rneq 5808	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → ran 𝑔 = ran ∅)
12	11	unieqd 4854	. . . . . . . . 9 ⊢ (𝑔 = ∅ → ∪ ran 𝑔 = ∪ ran ∅)
13		id 22	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → 𝑔 = ∅)
14	8	unieqd 4854	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → ∪ dom 𝑔 = ∪ dom ∅)
15	13, 14	fveq12d 6679	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → (𝑔‘∪ dom 𝑔) = (∅‘∪ dom ∅))
16	15	fveq2d 6676	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐹‘(∅‘∪ dom ∅)))
17	10, 12, 16	ifbieq12d 4496	. . . . . . . 8 ⊢ (𝑔 = ∅ → if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) = if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅))))
18	7, 17	ifbieq2d 4494	. . . . . . 7 ⊢ (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 2899	. . . . . 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 2823	. . . . . . 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)))))
21	20	dmmpt 6096	. . . . . 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}
22	19, 21	elrab2 3685	. . . . 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 2823	. . . . . . . . 9 ⊢ ∅ = ∅
24	23	iftruei 4476	. . . . . . . 8 ⊢ if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) = 𝐼
25	24	eleq1i 2905	. . . . . . 7 ⊢ (if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V ↔ 𝐼 ∈ V)
26	25	biimpi 218	. . . . . 6 ⊢ (if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V → 𝐼 ∈ V)
27	26	adantl 484	. . . . 5 ⊢ ((∅ ∈ V ∧ if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V) → 𝐼 ∈ V)
28	22, 27	sylbi 219	. . . 4 ⊢ (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) → 𝐼 ∈ V)
29	28	con3i 157	. . 3 ⊢ (¬ 𝐼 ∈ V → ¬ ∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
30		ndmfv 6702	. . 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)))))‘∅) = ∅)
31	29, 30	syl 17	. 2 ⊢ (¬ 𝐼 ∈ V → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘∅) = ∅)
32	6, 31	syl5eq 2870	1 ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ∅c0 4293  ifcif 4469  ∪ cuni 4840   ↦ cmpt 5148  dom cdm 5557  ran crn 5558   ↾ cres 5559  Oncon0 6193  Lim wlim 6194  ‘cfv 6357  reccrdg 8047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-wrecs 7949  df-recs 8010  df-rdg 8048

This theorem is referenced by:  rdgprc  33041