Theorem ordtypelem2 9434

Description: Lemma for ordtype 9447. (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 4022	. . . . . . . . 9 ⊢ 𝑇 ⊆ On
3	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ On)
4	3	sselda 3921	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ On)
5		onss 7739	. . . . . . 7 ⊢ (𝑎 ∈ On → 𝑎 ⊆ On)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ On)
7		eloni 6333	. . . . . . . 8 ⊢ (𝑎 ∈ On → Ord 𝑎)
8	4, 7	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → Ord 𝑎)
9		imaeq2 6021	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐹 “ 𝑥) = (𝐹 “ 𝑎))
10	9	raleqdv 3295	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
11	10	rexbidv 3161	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
12	11, 1	elrab2 3637	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝑇 ↔ (𝑎 ∈ On ∧ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
13	12	simprbi 497	. . . . . . . 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 6339	. . . . . . . . 9 ⊢ ((Ord 𝑎 ∧ 𝑥 ∈ 𝑎) → 𝑥 ⊆ 𝑎)
16		imass2 6067	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝑎 → (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎))
17		ssralv 3990	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎) → (∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
18	17	reximdv 3152	. . . . . . . . 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 3137	. . . . . . 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 4011	. . . . . 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 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡})
24	23, 1	sseqtrrdi 3963	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ 𝑇)
25	24	ralrimiva 3129	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇)
26		dftr3 5197	. . 3 ⊢ (Tr 𝑇 ↔ ∀𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇)
27	25, 26	sylibr 234	. 2 ⊢ (𝜑 → Tr 𝑇)
28		ordon 7731	. . 3 ⊢ Ord On
29		trssord 6340	. . 3 ⊢ ((Tr 𝑇 ∧ 𝑇 ⊆ On ∧ Ord On) → Ord 𝑇)
30	2, 28, 29	mp3an23 1456	. 2 ⊢ (Tr 𝑇 → Ord 𝑇)
31	27, 30	syl 17	1 ⊢ (𝜑 → Ord 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  {crab 3389  Vcvv 3429   ⊆ wss 3889   class class class wbr 5085   ↦ cmpt 5166  Tr wtr 5192   Se wse 5582   We wwe 5583  ran crn 5632   “ cima 5634  Ord word 6322  Oncon0 6323  ℩crio 7323  recscrecs 8310  OrdIsocoi 9424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327

This theorem is referenced by:  ordtypelem5  9437  ordtypelem6  9438  ordtypelem7  9439  ordtypelem8  9440  ordtypelem9  9441