Theorem rdg0 6596

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 4221	. . . . 5 ⊢ ∅ ∈ V
2		dmeq 4937	. . . . . . . 8 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
3		fveq1 5647	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔‘𝑥) = (∅‘𝑥))
4	3	fveq2d 5652	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘𝑥)) = (𝐹‘(∅‘𝑥)))
5	2, 4	iuneq12d 3999	. . . . . . 7 ⊢ (𝑔 = ∅ → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)))
6	5	uneq2d 3363	. . . . . 6 ⊢ (𝑔 = ∅ → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))))
7		eqid 2231	. . . . . 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 4951	. . . . . . . . . 10 ⊢ dom ∅ = ∅
10		iuneq1 3988	. . . . . . . . . 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 4033	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ ∅ (𝐹‘(∅‘𝑥)) = ∅
13	11, 12	eqtri 2252	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) = ∅
14	13, 1	eqeltri 2304	. . . . . . 7 ⊢ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥)) ∈ V
15	8, 14	unex 4544	. . . . . 6 ⊢ (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))) ∈ V
16	6, 7, 15	fvmpt 5732	. . . . 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 2304	. . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) ∈ V
19		df-irdg 6579	. . . 4 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
20	19	tfr0 6532	. . 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 3360	. . . 4 ⊢ (𝐴 ∪ ∪ 𝑥 ∈ dom ∅(𝐹‘(∅‘𝑥))) = (𝐴 ∪ ∅)
23	17, 22	eqtri 2252	. . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = (𝐴 ∪ ∅)
24		un0 3530	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
25	23, 24	eqtri 2252	. 2 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘∅) = 𝐴
26	21, 25	eqtri 2252	1 ⊢ (rec(𝐹, 𝐴)‘∅) = 𝐴

Syntax hints:   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ∪ cun 3199  ∅c0 3496  ∪ ciun 3975   ↦ cmpt 4155  dom cdm 4731  ‘cfv 5333  reccrdg 6578

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-recs 6514  df-irdg 6579

This theorem is referenced by:  rdg0g  6597  om0  6669