Theorem tfrlem8 8303

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 8297	. . . . . . . 8 ⊢ 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))}
3	2	eqabri 2874	. . . . . . 7 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
4		fndm 6584	. . . . . . . . . . 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 2816	. . . . . . . . 9 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → (dom 𝑔 ∈ On ↔ 𝑧 ∈ On))
7	6	biimprcd 250	. . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 ∈ On))
8	7	rexlimiv 3126	. . . . . . 7 ⊢ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 ∈ On)
9	3, 8	sylbi 217	. . . . . 6 ⊢ (𝑔 ∈ 𝐴 → dom 𝑔 ∈ On)
10		eleq1a 2826	. . . . . 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 3126	. . . 4 ⊢ (∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔 → 𝑧 ∈ On)
13	12	abssi 4015	. . 3 ⊢ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} ⊆ On
14		ssorduni 7712	. . 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 8300	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
17	16	dmeqi 5843	. . . 4 ⊢ dom recs(𝐹) = dom ∪ 𝐴
18		dmuni 5853	. . . 4 ⊢ dom ∪ 𝐴 = ∪ 𝑔 ∈ 𝐴 dom 𝑔
19		vex 3440	. . . . . 6 ⊢ 𝑔 ∈ V
20	19	dmex 7839	. . . . 5 ⊢ dom 𝑔 ∈ V
21	20	dfiun2 4980	. . . 4 ⊢ ∪ 𝑔 ∈ 𝐴 dom 𝑔 = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔}
22	17, 18, 21	3eqtri 2758	. . 3 ⊢ dom recs(𝐹) = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔}
23		ordeq 6313	. . 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 231	1 ⊢ Ord dom recs(𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  ∪ cuni 4856  ∪ ciun 4939  dom cdm 5614   ↾ cres 5616  Ord word 6305  Oncon0 6306   Fn wfn 6476  ‘cfv 6481  recscrecs 8290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fo 6487  df-fv 6489  df-ov 7349  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291

This theorem is referenced by:  tfrlem10  8306  tfrlem12  8308  tfrlem13  8309  tfrlem14  8310  tfrlem15  8311  tfrlem16  8312