Theorem ordtypelem2 9538

Description: Lemma for ordtype 9551. (Contributed by Mario Carneiro, 24-Jun-2015.)

Hypotheses

Ref	Expression
ordtypelem.1	⊢ 𝐹 = recs(𝐺)
ordtypelem.2	⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
ordtypelem.3	⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
ordtypelem.5	⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
ordtypelem.6	⊢ 𝑂 = OrdIso(𝑅, 𝐴)
ordtypelem.7	⊢ (𝜑 → 𝑅 We 𝐴)
ordtypelem.8	⊢ (𝜑 → 𝑅 Se 𝐴)

Assertion

Ref	Expression
ordtypelem2	⊢ (𝜑 → Ord 𝑇)

Distinct variable groups:   𝑣,𝑢,𝐶   ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑅   𝐴,ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   𝑡,𝑂,𝑢,𝑣,𝑥   𝜑,𝑡,𝑥   ℎ,𝐹,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑧,𝑤,𝑣,𝑢,ℎ,𝑗)   𝐶(𝑥,𝑧,𝑤,𝑡,ℎ,𝑗)   𝑇(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,ℎ,𝑗)   𝐺(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,ℎ,𝑗)   𝑂(𝑧,𝑤,ℎ,𝑗)

Proof of Theorem ordtypelem2

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtypelem.5	. . . . . . . . . 10 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
2	1	ssrab3 4062	. . . . . . . . 9 ⊢ 𝑇 ⊆ On
3	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ On)
4	3	sselda 3963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ On)
5		onss 7784	. . . . . . 7 ⊢ (𝑎 ∈ On → 𝑎 ⊆ On)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ On)
7		eloni 6367	. . . . . . . 8 ⊢ (𝑎 ∈ On → Ord 𝑎)
8	4, 7	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → Ord 𝑎)
9		imaeq2 6048	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐹 “ 𝑥) = (𝐹 “ 𝑎))
10	9	raleqdv 3309	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
11	10	rexbidv 3165	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
12	11, 1	elrab2 3679	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝑇 ↔ (𝑎 ∈ On ∧ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
13	12	simprbi 496	. . . . . . . 8 ⊢ (𝑎 ∈ 𝑇 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡)
14	13	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡)
15		ordelss 6373	. . . . . . . . 9 ⊢ ((Ord 𝑎 ∧ 𝑥 ∈ 𝑎) → 𝑥 ⊆ 𝑎)
16		imass2 6094	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝑎 → (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎))
17		ssralv 4032	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎) → (∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
18	17	reximdv 3156	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎) → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
19	15, 16, 18	3syl 18	. . . . . . . 8 ⊢ ((Ord 𝑎 ∧ 𝑥 ∈ 𝑎) → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
20	19	ralrimdva 3141	. . . . . . 7 ⊢ (Ord 𝑎 → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∀𝑥 ∈ 𝑎 ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
21	8, 14, 20	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∀𝑥 ∈ 𝑎 ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡)
22		ssrab 4053	. . . . . 6 ⊢ (𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} ↔ (𝑎 ⊆ On ∧ ∀𝑥 ∈ 𝑎 ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
23	6, 21, 22	sylanbrc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡})
24	23, 1	sseqtrrdi 4005	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ 𝑇)
25	24	ralrimiva 3133	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇)
26		dftr3 5240	. . 3 ⊢ (Tr 𝑇 ↔ ∀𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇)
27	25, 26	sylibr 234	. 2 ⊢ (𝜑 → Tr 𝑇)
28		ordon 7776	. . 3 ⊢ Ord On
29		trssord 6374	. . 3 ⊢ ((Tr 𝑇 ∧ 𝑇 ⊆ On ∧ Ord On) → Ord 𝑇)
30	2, 28, 29	mp3an23 1455	. 2 ⊢ (Tr 𝑇 → Ord 𝑇)
31	27, 30	syl 17	1 ⊢ (𝜑 → Ord 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  {crab 3420  Vcvv 3464   ⊆ wss 3931   class class class wbr 5124   ↦ cmpt 5206  Tr wtr 5234   Se wse 5609   We wwe 5610  ran crn 5660   “ cima 5662  Ord word 6356  Oncon0 6357  ℩crio 7366  recscrecs 8389  OrdIsocoi 9528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361

This theorem is referenced by:  ordtypelem5  9541  ordtypelem6  9542  ordtypelem7  9543  ordtypelem8  9544  ordtypelem9  9545