Theorem tfrlem8 7525

Description: Lemma for transfinite recursion. The domain of recs is an ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)

Hypothesis

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Assertion

Ref	Expression
tfrlem8	⊢ Ord dom recs(𝐹)

Distinct variable group:   𝑥,𝑓,𝑦,𝐹

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem8

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

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . . . . . . 9 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem3 7519	. . . . . . . 8 ⊢ 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))}
3	2	abeq2i 2764	. . . . . . 7 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
4		fndm 6028	. . . . . . . . . . 11 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
5	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 = 𝑧)
6	5	eleq1d 2715	. . . . . . . . 9 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → (dom 𝑔 ∈ On ↔ 𝑧 ∈ On))
7	6	biimprcd 240	. . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 ∈ On))
8	7	rexlimiv 3056	. . . . . . 7 ⊢ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 ∈ On)
9	3, 8	sylbi 207	. . . . . 6 ⊢ (𝑔 ∈ 𝐴 → dom 𝑔 ∈ On)
10		eleq1a 2725	. . . . . 6 ⊢ (dom 𝑔 ∈ On → (𝑧 = dom 𝑔 → 𝑧 ∈ On))
11	9, 10	syl 17	. . . . 5 ⊢ (𝑔 ∈ 𝐴 → (𝑧 = dom 𝑔 → 𝑧 ∈ On))
12	11	rexlimiv 3056	. . . 4 ⊢ (∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔 → 𝑧 ∈ On)
13	12	abssi 3710	. . 3 ⊢ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} ⊆ On
14		ssorduni 7027	. . 3 ⊢ ({𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} ⊆ On → Ord ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔})
15	13, 14	ax-mp 5	. 2 ⊢ Ord ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔}
16	1	recsfval 7522	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
17	16	dmeqi 5357	. . . 4 ⊢ dom recs(𝐹) = dom ∪ 𝐴
18		dmuni 5366	. . . 4 ⊢ dom ∪ 𝐴 = ∪ 𝑔 ∈ 𝐴 dom 𝑔
19		vex 3234	. . . . . 6 ⊢ 𝑔 ∈ V
20	19	dmex 7141	. . . . 5 ⊢ dom 𝑔 ∈ V
21	20	dfiun2 4586	. . . 4 ⊢ ∪ 𝑔 ∈ 𝐴 dom 𝑔 = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔}
22	17, 18, 21	3eqtri 2677	. . 3 ⊢ dom recs(𝐹) = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔}
23		ordeq 5768	. . 3 ⊢ (dom recs(𝐹) = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} → (Ord dom recs(𝐹) ↔ Ord ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔}))
24	22, 23	ax-mp 5	. 2 ⊢ (Ord dom recs(𝐹) ↔ Ord ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔})
25	15, 24	mpbir 221	1 ⊢ Ord dom recs(𝐹)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637  ∀wral 2941  ∃wrex 2942   ⊆ wss 3607  ∪ cuni 4468  ∪ ciun 4552  dom cdm 5143   ↾ cres 5145  Ord word 5760  Oncon0 5761   Fn wfn 5921  ‘cfv 5926  recscrecs 7512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-wrecs 7452  df-recs 7513

This theorem is referenced by:  tfrlem10  7528  tfrlem12  7530  tfrlem13  7531  tfrlem14  7532  tfrlem15  7533  tfrlem16  7534