Theorem rdgon 5996

Description: Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.)

Hypotheses

Ref	Expression
rdgon.1	⊢ (𝜑 → 𝐹 Fn V)
rdgon.2	⊢ (𝜑 → 𝐴 ∈ On)
rdgon.3	⊢ (𝜑 → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)

Assertion

Ref	Expression
rdgon	⊢ ((𝜑 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝜑,𝑥

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rdgon

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5198	. . . . 5 ⊢ (𝑧 = 𝑥 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘𝑥))
2	1	eleq1d 2147	. . . 4 ⊢ (𝑧 = 𝑥 → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
3	2	imbi2d 228	. . 3 ⊢ (𝑧 = 𝑥 → ((𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On) ↔ (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On)))
4		fveq2 5198	. . . . 5 ⊢ (𝑧 = 𝐵 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘𝐵))
5	4	eleq1d 2147	. . . 4 ⊢ (𝑧 = 𝐵 → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (rec(𝐹, 𝐴)‘𝐵) ∈ On))
6	5	imbi2d 228	. . 3 ⊢ (𝑧 = 𝐵 → ((𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On) ↔ (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ∈ On)))
7		r19.21v 2438	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On) ↔ (𝜑 → ∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On))
8		rdgon.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ On)
9		fvres 5219	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑧 → ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) = (rec(𝐹, 𝐴)‘𝑥))
10	9	eleq1d 2147	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑧 → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
11	10	adantl 271	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑧) → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
12		rdgon.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)
13		fveq2 5198	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
14	13	eleq1d 2147	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ∈ On ↔ (𝐹‘𝑤) ∈ On))
15	14	cbvralv 2577	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ On ↔ ∀𝑤 ∈ On (𝐹‘𝑤) ∈ On)
16	12, 15	sylib 120	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑤 ∈ On (𝐹‘𝑤) ∈ On)
17		fveq2 5198	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) → (𝐹‘𝑤) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)))
18	17	eleq1d 2147	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) → ((𝐹‘𝑤) ∈ On ↔ (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
19	18	rspcv 2697	. . . . . . . . . . . . . 14 ⊢ (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (∀𝑤 ∈ On (𝐹‘𝑤) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
20	16, 19	syl5com 29	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
21	20	adantr 270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑧) → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
22	11, 21	sylbird 168	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑧) → ((rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
23	22	ralimdva 2429	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → ∀𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
24		vex 2604	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
25		iunon 5922	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ ∀𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On) → ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On)
26	24, 25	mpan 414	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On → ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On)
27	23, 26	syl6 33	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
28		onun2 4234	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On) → (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On)
29	8, 27, 28	syl6an 1363	. . . . . . . 8 ⊢ (𝜑 → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
30	29	adantr 270	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
31		rdgon.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn V)
32	31, 8	jca 300	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 Fn V ∧ 𝐴 ∈ On))
33		rdgivallem 5991	. . . . . . . . . 10 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ On ∧ 𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
34	33	3expa 1138	. . . . . . . . 9 ⊢ (((𝐹 Fn V ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
35	32, 34	sylan 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
36	35	eleq1d 2147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
37	30, 36	sylibrd 167	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (rec(𝐹, 𝐴)‘𝑧) ∈ On))
38	37	expcom 114	. . . . 5 ⊢ (𝑧 ∈ On → (𝜑 → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
39	38	a2d 26	. . . 4 ⊢ (𝑧 ∈ On → ((𝜑 → ∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On) → (𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
40	7, 39	syl5bi 150	. . 3 ⊢ (𝑧 ∈ On → (∀𝑥 ∈ 𝑧 (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On) → (𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
41	3, 6, 40	tfis3 4327	. 2 ⊢ (𝐵 ∈ On → (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ∈ On))
42	41	impcom 123	1 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348  Vcvv 2601   ∪ cun 2971  ∪ ciun 3678  Oncon0 4118   ↾ cres 4365   Fn wfn 4917  ‘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-coll 3893  ax-sep 3896  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-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  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-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943  df-irdg 5980

This theorem is referenced by:  oacl  6063  omcl  6064  oeicl  6065