Theorem rdgon 6386

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.) (Revised by Jim Kingdon, 13-Apr-2022.)

Hypotheses

Ref	Expression
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,𝐵   𝑥,𝐹   𝜑,𝑥

Proof of Theorem rdgon

Dummy variables 𝑓 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-irdg 6370	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
2		funmpt 5254	. . 3 ⊢ Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))
3	2	a1i 9	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
4		ordon 4485	. . 3 ⊢ Ord On
5	4	a1i 9	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → Ord On)
6		vex 2740	. . . 4 ⊢ 𝑓 ∈ V
7		rdgon.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
8	7	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
9	8	3ad2ant1 1018	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → 𝐴 ∈ On)
10	6	dmex 4893	. . . . . 6 ⊢ dom 𝑓 ∈ V
11		fveq2 5515	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘𝑧) → (𝐹‘𝑥) = (𝐹‘(𝑓‘𝑧)))
12	11	eleq1d 2246	. . . . . . . . 9 ⊢ (𝑥 = (𝑓‘𝑧) → ((𝐹‘𝑥) ∈ On ↔ (𝐹‘(𝑓‘𝑧)) ∈ On))
13		rdgon.3	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)
14	13	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)
15	14	3ad2ant1 1018	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)
16	15	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)
17		simpl3 1002	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → 𝑓:𝑦⟶On)
18		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → 𝑧 ∈ dom 𝑓)
19		fdm 5371	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑦⟶On → dom 𝑓 = 𝑦)
20	19	eleq2d 2247	. . . . . . . . . . . 12 ⊢ (𝑓:𝑦⟶On → (𝑧 ∈ dom 𝑓 ↔ 𝑧 ∈ 𝑦))
21	17, 20	syl 14	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → (𝑧 ∈ dom 𝑓 ↔ 𝑧 ∈ 𝑦))
22	18, 21	mpbid 147	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → 𝑧 ∈ 𝑦)
23	17, 22	ffvelcdmd 5652	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → (𝑓‘𝑧) ∈ On)
24	12, 16, 23	rspcdva 2846	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → (𝐹‘(𝑓‘𝑧)) ∈ On)
25	24	ralrimiva 2550	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ∀𝑧 ∈ dom 𝑓(𝐹‘(𝑓‘𝑧)) ∈ On)
26		fveq2 5515	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑓‘𝑥) = (𝑓‘𝑧))
27	26	fveq2d 5519	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑧)))
28	27	eleq1d 2246	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝑓‘𝑥)) ∈ On ↔ (𝐹‘(𝑓‘𝑧)) ∈ On))
29	28	cbvralv 2703	. . . . . . 7 ⊢ (∀𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On ↔ ∀𝑧 ∈ dom 𝑓(𝐹‘(𝑓‘𝑧)) ∈ On)
30	25, 29	sylibr 134	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ∀𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On)
31		iunon 6284	. . . . . 6 ⊢ ((dom 𝑓 ∈ V ∧ ∀𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On) → ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On)
32	10, 30, 31	sylancr 414	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On)
33		onun2 4489	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))) ∈ On)
34	9, 32, 33	syl2anc 411	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))) ∈ On)
35		dmeq 4827	. . . . . . 7 ⊢ (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
36		fveq1 5514	. . . . . . . 8 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑥) = (𝑓‘𝑥))
37	36	fveq2d 5519	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝐹‘(𝑔‘𝑥)) = (𝐹‘(𝑓‘𝑥)))
38	35, 37	iuneq12d 3910	. . . . . 6 ⊢ (𝑔 = 𝑓 → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)))
39	38	uneq2d 3289	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))))
40		eqid 2177	. . . . 5 ⊢ (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))
41	39, 40	fvmptg 5592	. . . 4 ⊢ ((𝑓 ∈ V ∧ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))) ∈ On) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))))
42	6, 34, 41	sylancr 414	. . 3 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))))
43	42, 34	eqeltrd 2254	. 2 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ On)
44		unon 4510	. . . . . 6 ⊢ ∪ On = On
45	44	eleq2i 2244	. . . . 5 ⊢ (𝑦 ∈ ∪ On ↔ 𝑦 ∈ On)
46	45	biimpi 120	. . . 4 ⊢ (𝑦 ∈ ∪ On → 𝑦 ∈ On)
47	46	adantl 277	. . 3 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ∪ On) → 𝑦 ∈ On)
48		onsuc 4500	. . 3 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
49	47, 48	syl 14	. 2 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ∪ On) → suc 𝑦 ∈ On)
50	44	eleq2i 2244	. . . 4 ⊢ (𝐵 ∈ ∪ On ↔ 𝐵 ∈ On)
51	50	biimpri 133	. . 3 ⊢ (𝐵 ∈ On → 𝐵 ∈ ∪ On)
52	51	adantl 277	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → 𝐵 ∈ ∪ On)
53	1, 3, 5, 43, 49, 52	tfrcl 6364	1 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  Vcvv 2737   ∪ cun 3127  ∪ cuni 3809  ∪ ciun 3886   ↦ cmpt 4064  Ord word 4362  Oncon0 4363  suc csuc 4365  dom cdm 4626  Fun wfun 5210  ⟶wf 5212  ‘cfv 5216  reccrdg 6369

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-recs 6305  df-irdg 6370

This theorem is referenced by:  oacl  6460  omcl  6461  oeicl  6462