Theorem rdgon 6630

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 6614	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
2		funmpt 5395	. . 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 4613	. . 3 ⊢ Ord On
5	4	a1i 9	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → Ord On)
6		vex 2818	. . . 4 ⊢ 𝑓 ∈ V
7		rdgon.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
8	7	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
9	8	3ad2ant1 1045	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → 𝐴 ∈ On)
10	6	dmex 5029	. . . . . 6 ⊢ dom 𝑓 ∈ V
11		fveq2 5675	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘𝑧) → (𝐹‘𝑥) = (𝐹‘(𝑓‘𝑧)))
12	11	eleq1d 2303	. . . . . . . . 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 1045	. . . . . . . . . 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 1029	. . . . . . . . . 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 5519	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑦⟶On → dom 𝑓 = 𝑦)
20	19	eleq2d 2304	. . . . . . . . . . . 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 5818	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → (𝑓‘𝑧) ∈ On)
24	12, 16, 23	rspcdva 2928	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → (𝐹‘(𝑓‘𝑧)) ∈ On)
25	24	ralrimiva 2617	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ∀𝑧 ∈ dom 𝑓(𝐹‘(𝑓‘𝑧)) ∈ On)
26		fveq2 5675	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑓‘𝑥) = (𝑓‘𝑧))
27	26	fveq2d 5679	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑧)))
28	27	eleq1d 2303	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝑓‘𝑥)) ∈ On ↔ (𝐹‘(𝑓‘𝑧)) ∈ On))
29	28	cbvralv 2780	. . . . . . 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 6528	. . . . . 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 4617	. . . . 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 4961	. . . . . . 7 ⊢ (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
36		fveq1 5674	. . . . . . . 8 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑥) = (𝑓‘𝑥))
37	36	fveq2d 5679	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝐹‘(𝑔‘𝑥)) = (𝐹‘(𝑓‘𝑥)))
38	35, 37	iuneq12d 4020	. . . . . 6 ⊢ (𝑔 = 𝑓 → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)))
39	38	uneq2d 3377	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))))
40		eqid 2234	. . . . 5 ⊢ (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))
41	39, 40	fvmptg 5758	. . . 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 2311	. 2 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ On)
44		unon 4638	. . . . . 6 ⊢ ∪ On = On
45	44	eleq2i 2301	. . . . 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 4628	. . 3 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
49	47, 48	syl 14	. 2 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ∪ On) → suc 𝑦 ∈ On)
50	44	eleq2i 2301	. . . 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 6608	1 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815   ∪ cun 3212  ∪ cuni 3919  ∪ ciun 3996   ↦ cmpt 4176  Ord word 4488  Oncon0 4489  suc csuc 4491  dom cdm 4754  Fun wfun 5351  ⟶wf 5353  ‘cfv 5357  reccrdg 6613

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-recs 6549  df-irdg 6614

This theorem is referenced by:  oacl  6706  omcl  6707  oeicl  6708