Theorem rdg0 6503

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 4190	. . . . 5 ⊢ ∅ ∈ V
2		dmeq 4900	. . . . . . . 8 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
3		fveq1 5602	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔‘𝑥) = (∅‘𝑥))
4	3	fveq2d 5607	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘𝑥)) = (𝐹‘(∅‘𝑥)))
5	2, 4	iuneq12d 3968	. . . . . . 7 ⊢ (𝑔 = ∅ → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)))
6	5	uneq2d 3338	. . . . . 6 ⊢ (𝑔 = ∅ → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))))
7		eqid 2209	. . . . . 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 4914	. . . . . . . . . 10 ⊢ dom ∅ = ∅
10		iuneq1 3957	. . . . . . . . . 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 4002	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ ∅ (𝐹‘(∅‘𝑥)) = ∅
13	11, 12	eqtri 2230	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) = ∅
14	13, 1	eqeltri 2282	. . . . . . 7 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) ∈ V
15	8, 14	unex 4509	. . . . . 6 ⊢ (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))) ∈ V
16	6, 7, 15	fvmpt 5684	. . . . 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 2282	. . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) ∈ V
19		df-irdg 6486	. . . 4 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
20	19	tfr0 6439	. . 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 3335	. . . 4 ⊢ (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))) = (𝐴 ∪ ∅)
23	17, 22	eqtri 2230	. . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = (𝐴 ∪ ∅)
24		un0 3505	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
25	23, 24	eqtri 2230	. 2 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = 𝐴
26	21, 25	eqtri 2230	1 ⊢ (rec(𝐹, 𝐴)‘∅) = 𝐴

Syntax hints:   = wceq 1375   ∈ wcel 2180  Vcvv 2779   ∪ cun 3175  ∅c0 3471  ∪ ciun 3944   ↦ cmpt 4124  dom cdm 4696  ‘cfv 5294  reccrdg 6485

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-res 4708  df-iota 5254  df-fun 5296  df-fn 5297  df-fv 5302  df-recs 6421  df-irdg 6486

This theorem is referenced by:  rdg0g  6504  om0  6574