Theorem ordtypelem2 9557

Description: Lemma for ordtype 9570. (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 4092	. . . . . . . . 9 ⊢ 𝑇 ⊆ On
3	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ On)
4	3	sselda 3995	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ On)
5		onss 7804	. . . . . . 7 ⊢ (𝑎 ∈ On → 𝑎 ⊆ On)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ On)
7		eloni 6396	. . . . . . . 8 ⊢ (𝑎 ∈ On → Ord 𝑎)
8	4, 7	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → Ord 𝑎)
9		imaeq2 6076	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐹 “ 𝑥) = (𝐹 “ 𝑎))
10	9	raleqdv 3324	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
11	10	rexbidv 3177	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
12	11, 1	elrab2 3698	. . . . . . . . 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 6402	. . . . . . . . 9 ⊢ ((Ord 𝑎 ∧ 𝑥 ∈ 𝑎) → 𝑥 ⊆ 𝑎)
16		imass2 6123	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝑎 → (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎))
17		ssralv 4064	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎) → (∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
18	17	reximdv 3168	. . . . . . . . 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 3152	. . . . . . 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 4083	. . . . . 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 4047	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ 𝑇)
25	24	ralrimiva 3144	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇)
26		dftr3 5271	. . 3 ⊢ (Tr 𝑇 ↔ ∀𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇)
27	25, 26	sylibr 234	. 2 ⊢ (𝜑 → Tr 𝑇)
28		ordon 7796	. . 3 ⊢ Ord On
29		trssord 6403	. . 3 ⊢ ((Tr 𝑇 ∧ 𝑇 ⊆ On ∧ Ord On) → Ord 𝑇)
30	2, 28, 29	mp3an23 1452	. 2 ⊢ (Tr 𝑇 → Ord 𝑇)
31	27, 30	syl 17	1 ⊢ (𝜑 → Ord 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  {crab 3433  Vcvv 3478   ⊆ wss 3963   class class class wbr 5148   ↦ cmpt 5231  Tr wtr 5265   Se wse 5639   We wwe 5640  ran crn 5690   “ cima 5692  Ord word 6385  Oncon0 6386  ℩crio 7387  recscrecs 8409  OrdIsocoi 9547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390

This theorem is referenced by:  ordtypelem5  9560  ordtypelem6  9561  ordtypelem7  9562  ordtypelem8  9563  ordtypelem9  9564