Theorem ordtypelem2 8424

Description: Lemma for ordtype 8437. (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		ssrab2 3687	. . . . . . . . . 10 ⊢ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} ⊆ On
3	1, 2	eqsstri 3635	. . . . . . . . 9 ⊢ 𝑇 ⊆ On
4	3	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ On)
5	4	sselda 3603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ On)
6		onss 6990	. . . . . . 7 ⊢ (𝑎 ∈ On → 𝑎 ⊆ On)
7	5, 6	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ On)
8		eloni 5733	. . . . . . . 8 ⊢ (𝑎 ∈ On → Ord 𝑎)
9	5, 8	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → Ord 𝑎)
10		imaeq2 5462	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐹 “ 𝑥) = (𝐹 “ 𝑎))
11	10	raleqdv 3144	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
12	11	rexbidv 3052	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
13	12, 1	elrab2 3366	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝑇 ↔ (𝑎 ∈ On ∧ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
14	13	simprbi 480	. . . . . . . 8 ⊢ (𝑎 ∈ 𝑇 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡)
15	14	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡)
16		ordelss 5739	. . . . . . . . 9 ⊢ ((Ord 𝑎 ∧ 𝑥 ∈ 𝑎) → 𝑥 ⊆ 𝑎)
17		imass2 5501	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝑎 → (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎))
18		ssralv 3666	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎) → (∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
19	18	reximdv 3016	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎) → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
20	16, 17, 19	3syl 18	. . . . . . . 8 ⊢ ((Ord 𝑎 ∧ 𝑥 ∈ 𝑎) → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
21	20	ralrimdva 2969	. . . . . . 7 ⊢ (Ord 𝑎 → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∀𝑥 ∈ 𝑎 ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
22	9, 15, 21	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∀𝑥 ∈ 𝑎 ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡)
23		ssrab 3680	. . . . . 6 ⊢ (𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} ↔ (𝑎 ⊆ On ∧ ∀𝑥 ∈ 𝑎 ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
24	7, 22, 23	sylanbrc 698	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡})
25	24, 1	syl6sseqr 3652	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ 𝑇)
26	25	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇)
27		dftr3 4756	. . 3 ⊢ (Tr 𝑇 ↔ ∀𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇)
28	26, 27	sylibr 224	. 2 ⊢ (𝜑 → Tr 𝑇)
29		ordon 6982	. . 3 ⊢ Ord On
30		trssord 5740	. . 3 ⊢ ((Tr 𝑇 ∧ 𝑇 ⊆ On ∧ Ord On) → Ord 𝑇)
31	3, 29, 30	mp3an23 1416	. 2 ⊢ (Tr 𝑇 → Ord 𝑇)
32	28, 31	syl 17	1 ⊢ (𝜑 → Ord 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ⊆ wss 3574   class class class wbr 4653   ↦ cmpt 4729  Tr wtr 4752   Se wse 5071   We wwe 5072  ran crn 5115   “ cima 5117  Ord word 5722  Oncon0 5723  ℩crio 6610  recscrecs 7467  OrdIsocoi 8414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727

This theorem is referenced by:  ordtypelem5  8427  ordtypelem6  8428  ordtypelem7  8429  ordtypelem8  8430  ordtypelem9  8431