Theorem rdg0 6402

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 4142	. . . . 5 ⊢ ∅ ∈ V
2		dmeq 4839	. . . . . . . 8 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
3		fveq1 5526	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔‘𝑥) = (∅‘𝑥))
4	3	fveq2d 5531	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘𝑥)) = (𝐹‘(∅‘𝑥)))
5	2, 4	iuneq12d 3922	. . . . . . 7 ⊢ (𝑔 = ∅ → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)))
6	5	uneq2d 3301	. . . . . 6 ⊢ (𝑔 = ∅ → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))))
7		eqid 2187	. . . . . 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 4853	. . . . . . . . . 10 ⊢ dom ∅ = ∅
10		iuneq1 3911	. . . . . . . . . 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 3956	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ ∅ (𝐹‘(∅‘𝑥)) = ∅
13	11, 12	eqtri 2208	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) = ∅
14	13, 1	eqeltri 2260	. . . . . . 7 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) ∈ V
15	8, 14	unex 4453	. . . . . 6 ⊢ (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))) ∈ V
16	6, 7, 15	fvmpt 5606	. . . . 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 2260	. . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) ∈ V
19		df-irdg 6385	. . . 4 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
20	19	tfr0 6338	. . 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 3298	. . . 4 ⊢ (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))) = (𝐴 ∪ ∅)
23	17, 22	eqtri 2208	. . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = (𝐴 ∪ ∅)
24		un0 3468	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
25	23, 24	eqtri 2208	. 2 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = 𝐴
26	21, 25	eqtri 2208	1 ⊢ (rec(𝐹, 𝐴)‘∅) = 𝐴

Syntax hints:   = wceq 1363   ∈ wcel 2158  Vcvv 2749   ∪ cun 3139  ∅c0 3434  ∪ ciun 3898   ↦ cmpt 4076  dom cdm 4638  ‘cfv 5228  reccrdg 6384

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-recs 6320  df-irdg 6385

This theorem is referenced by:  rdg0g  6403  om0  6473