Theorem rdg0 5997

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 3905	. . . . 5 ⊢ ∅ ∈ V
2		dmeq 4553	. . . . . . . 8 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
3		fveq1 5197	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔‘𝑥) = (∅‘𝑥))
4	3	fveq2d 5202	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘𝑥)) = (𝐹‘(∅‘𝑥)))
5	2, 4	iuneq12d 3702	. . . . . . 7 ⊢ (𝑔 = ∅ → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)))
6	5	uneq2d 3126	. . . . . 6 ⊢ (𝑔 = ∅ → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))))
7		eqid 2081	. . . . . 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 4567	. . . . . . . . . 10 ⊢ dom ∅ = ∅
10		iuneq1 3691	. . . . . . . . . 10 ⊢ (dom ∅ = ∅ → ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) = ∪ 𝑥 ∈ ∅ (𝐹‘(∅‘𝑥)))
11	9, 10	ax-mp 7	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) = ∪ 𝑥 ∈ ∅ (𝐹‘(∅‘𝑥))
12		0iun 3735	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ ∅ (𝐹‘(∅‘𝑥)) = ∅
13	11, 12	eqtri 2101	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) = ∅
14	13, 1	eqeltri 2151	. . . . . . 7 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) ∈ V
15	8, 14	unex 4194	. . . . . 6 ⊢ (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))) ∈ V
16	6, 7, 15	fvmpt 5270	. . . . 5 ⊢ (∅ ∈ V → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))))
17	1, 16	ax-mp 7	. . . 4 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)))
18	17, 15	eqeltri 2151	. . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) ∈ V
19		df-irdg 5980	. . . 4 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
20	19	tfr0 5960	. . 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 7	. 2 ⊢ (rec(𝐹, 𝐴)‘∅) = ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅)
22	13	uneq2i 3123	. . . 4 ⊢ (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))) = (𝐴 ∪ ∅)
23	17, 22	eqtri 2101	. . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = (𝐴 ∪ ∅)
24		un0 3278	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
25	23, 24	eqtri 2101	. 2 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = 𝐴
26	21, 25	eqtri 2101	1 ⊢ (rec(𝐹, 𝐴)‘∅) = 𝐴

Syntax hints:   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ∪ cun 2971  ∅c0 3251  ∪ ciun 3678   ↦ cmpt 3839  dom cdm 4363  ‘cfv 4922  reccrdg 5979

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930  df-recs 5943  df-irdg 5980

This theorem is referenced by:  rdg0g  5998  om0  6061