Theorem rdg0 6445

Description: The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Hypothesis

Ref	Expression
rdg.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
rdg0	⊢ (rec(𝐹, 𝐴)‘∅) = 𝐴

Proof of Theorem rdg0

Dummy variables 𝑥 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4160	. . . . 5 ⊢ ∅ ∈ V
2		dmeq 4866	. . . . . . . 8 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
3		fveq1 5557	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔‘𝑥) = (∅‘𝑥))
4	3	fveq2d 5562	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘𝑥)) = (𝐹‘(∅‘𝑥)))
5	2, 4	iuneq12d 3940	. . . . . . 7 ⊢ (𝑔 = ∅ → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)))
6	5	uneq2d 3317	. . . . . 6 ⊢ (𝑔 = ∅ → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))))
7		eqid 2196	. . . . . 6 ⊢ (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))
8		rdg.1	. . . . . . 7 ⊢ 𝐴 ∈ V
9		dm0 4880	. . . . . . . . . 10 ⊢ dom ∅ = ∅
10		iuneq1 3929	. . . . . . . . . 10 ⊢ (dom ∅ = ∅ → ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) = ∪ 𝑥 ∈ ∅ (𝐹‘(∅‘𝑥)))
11	9, 10	ax-mp 5	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) = ∪ 𝑥 ∈ ∅ (𝐹‘(∅‘𝑥))
12		0iun 3974	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ ∅ (𝐹‘(∅‘𝑥)) = ∅
13	11, 12	eqtri 2217	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) = ∅
14	13, 1	eqeltri 2269	. . . . . . 7 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) ∈ V
15	8, 14	unex 4476	. . . . . 6 ⊢ (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))) ∈ V
16	6, 7, 15	fvmpt 5638	. . . . 5 ⊢ (∅ ∈ V → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))))
17	1, 16	ax-mp 5	. . . 4 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)))
18	17, 15	eqeltri 2269	. . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) ∈ V
19		df-irdg 6428	. . . 4 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
20	19	tfr0 6381	. . 3 ⊢ (((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) ∈ V → (rec(𝐹, 𝐴)‘∅) = ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅))
21	18, 20	ax-mp 5	. 2 ⊢ (rec(𝐹, 𝐴)‘∅) = ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅)
22	13	uneq2i 3314	. . . 4 ⊢ (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))) = (𝐴 ∪ ∅)
23	17, 22	eqtri 2217	. . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = (𝐴 ∪ ∅)
24		un0 3484	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
25	23, 24	eqtri 2217	. 2 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = 𝐴
26	21, 25	eqtri 2217	1 ⊢ (rec(𝐹, 𝐴)‘∅) = 𝐴

Syntax hints:   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∪ cun 3155  ∅c0 3450  ∪ ciun 3916   ↦ cmpt 4094  dom cdm 4663  ‘cfv 5258  reccrdg 6427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-recs 6363  df-irdg 6428

This theorem is referenced by:  rdg0g  6446  om0  6516