Theorem tfrlem8 5875

Description: Lemma for transfinite recursion. The domain of recs is 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 5867	. . . . . . . 8 ⊢ A = {g ∣ ∃z ∈ On (g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w)))}
3	2	abeq2i 2145	. . . . . . 7 ⊢ (g ∈ A ↔ ∃z ∈ On (g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))))
4		fndm 4941	. . . . . . . . . . 11 ⊢ (g Fn z → dom g = z)
5	4	adantr 261	. . . . . . . . . 10 ⊢ ((g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))) → dom g = z)
6	5	eleq1d 2103	. . . . . . . . 9 ⊢ ((g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))) → (dom g ∈ On ↔ z ∈ On))
7	6	biimprcd 149	. . . . . . . 8 ⊢ (z ∈ On → ((g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))) → dom g ∈ On))
8	7	rexlimiv 2421	. . . . . . 7 ⊢ (∃z ∈ On (g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))) → dom g ∈ On)
9	3, 8	sylbi 114	. . . . . 6 ⊢ (g ∈ A → dom g ∈ On)
10		eleq1a 2106	. . . . . 6 ⊢ (dom g ∈ On → (z = dom g → z ∈ On))
11	9, 10	syl 14	. . . . 5 ⊢ (g ∈ A → (z = dom g → z ∈ On))
12	11	rexlimiv 2421	. . . 4 ⊢ (∃g ∈ A z = dom g → z ∈ On)
13	12	abssi 3009	. . 3 ⊢ {z ∣ ∃g ∈ A z = dom g} ⊆ On
14		ssorduni 4179	. . 3 ⊢ ({z ∣ ∃g ∈ A z = dom g} ⊆ On → Ord ∪ {z ∣ ∃g ∈ A z = dom g})
15	13, 14	ax-mp 7	. 2 ⊢ Ord ∪ {z ∣ ∃g ∈ A z = dom g}
16	1	recsfval 5872	. . . . 5 ⊢ recs(𝐹) = ∪ A
17	16	dmeqi 4479	. . . 4 ⊢ dom recs(𝐹) = dom ∪ A
18		dmuni 4488	. . . 4 ⊢ dom ∪ A = ∪ g ∈ A dom g
19		vex 2554	. . . . . 6 ⊢ g ∈ V
20	19	dmex 4541	. . . . 5 ⊢ dom g ∈ V
21	20	dfiun2 3682	. . . 4 ⊢ ∪ g ∈ A dom g = ∪ {z ∣ ∃g ∈ A z = dom g}
22	17, 18, 21	3eqtri 2061	. . 3 ⊢ dom recs(𝐹) = ∪ {z ∣ ∃g ∈ A z = dom g}
23		ordeq 4075	. . 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 7	. 2 ⊢ (Ord dom recs(𝐹) ↔ Ord ∪ {z ∣ ∃g ∈ A z = dom g})
25	15, 24	mpbir 134	1 ⊢ Ord dom recs(𝐹)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301   ⊆ wss 2911  ∪ cuni 3571  ∪ ciun 3648  Ord word 4065  Oncon0 4066  dom cdm 4288   ↾ cres 4290   Fn wfn 4840  ‘cfv 4845  recscrecs 5860

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-tr 3846  df-iord 4069  df-on 4071  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-recs 5861

This theorem is referenced by:  tfrlemi14d  5888