Theorem tfrlem8 8355

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 8348	. . . . . . . 8 ⊢ 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))}
3	2	eqabri 2904	. . . . . . 7 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
4		fndm 6624	. . . . . . . . . . 11 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
5	4	adantr 484	. . . . . . . . . 10 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 = 𝑧)
6	5	eleq1d 2847	. . . . . . . . 9 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → (dom 𝑔 ∈ On ↔ 𝑧 ∈ On))
7	6	biimprcd 252	. . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 ∈ On))
8	7	rexlimiv 3156	. . . . . . 7 ⊢ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 ∈ On)
9	3, 8	sylbi 219	. . . . . 6 ⊢ (𝑔 ∈ 𝐴 → dom 𝑔 ∈ On)
10		eleq1a 2857	. . . . . 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 3156	. . . 4 ⊢ (∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔 → 𝑧 ∈ On)
13	12	abssi 4021	. . 3 ⊢ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} ⊆ On
14		ssorduni 7762	. . 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 8351	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
17	16	dmeqi 5880	. . . 4 ⊢ dom recs(𝐹) = dom ∪ 𝐴
18		dmuni 5890	. . . 4 ⊢ dom ∪ 𝐴 = ∪ 𝑔 ∈ 𝐴 dom 𝑔
19		vex 3458	. . . . . 6 ⊢ 𝑔 ∈ V
20	19	dmex 7890	. . . . 5 ⊢ dom 𝑔 ∈ V
21	20	dfiun2 4989	. . . 4 ⊢ ∪ 𝑔 ∈ 𝐴 dom 𝑔 = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔}
22	17, 18, 21	3eqtri 2789	. . 3 ⊢ dom recs(𝐹) = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔}
23		ordeq 6353	. . 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 233	1 ⊢ Ord dom recs(𝐹)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {cab 2740  ∀wral 3076  ∃wrex 3086   ⊆ wss 3904  ∪ cuni 4865  ∪ ciun 4949  dom cdm 5647   ↾ cres 5649  Ord word 6345  Oncon0 6346   Fn wfn 6516  ‘cfv 6521  recscrecs 8341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fo 6527  df-fv 6529  df-ov 7399  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342

This theorem is referenced by:  tfrlem10  8358  tfrlem12  8360  tfrlem13  8361  tfrlem14  8362  tfrlem15  8363  tfrlem16  8364