Theorem ordtypelem6 9482

Description: Lemma for ordtype 9491. (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
ordtypelem6	⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (𝑁 ∈ 𝑀 → (𝑂‘𝑁)𝑅(𝑂‘𝑀)))

Distinct variable groups:   𝑣,𝑢,𝐶   ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑀   𝑗,𝑁,𝑢,𝑤   𝑅,ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   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)   𝑁(𝑥,𝑧,𝑣,𝑡,ℎ)   𝑂(𝑧,𝑤,ℎ,𝑗)

Proof of Theorem ordtypelem6

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6860	. . . . 5 ⊢ (𝑎 = 𝑁 → (𝐹‘𝑎) = (𝐹‘𝑁))
2	1	breq1d 5119	. . . 4 ⊢ (𝑎 = 𝑁 → ((𝐹‘𝑎)𝑅(𝐹‘𝑀) ↔ (𝐹‘𝑁)𝑅(𝐹‘𝑀)))
3		ssrab2 4045	. . . . . . . 8 ⊢ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}
4		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ dom 𝑂)
5		ordtypelem.1	. . . . . . . . . . . . 13 ⊢ 𝐹 = recs(𝐺)
6		ordtypelem.2	. . . . . . . . . . . . 13 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
7		ordtypelem.3	. . . . . . . . . . . . 13 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
8		ordtypelem.5	. . . . . . . . . . . . 13 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
9		ordtypelem.6	. . . . . . . . . . . . 13 ⊢ 𝑂 = OrdIso(𝑅, 𝐴)
10		ordtypelem.7	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 We 𝐴)
11		ordtypelem.8	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 Se 𝐴)
12	5, 6, 7, 8, 9, 10, 11	ordtypelem4 9480	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
13	12	fdmd 6700	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
14	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → dom 𝑂 = (𝑇 ∩ dom 𝐹))
15	4, 14	eleqtrd 2831	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ (𝑇 ∩ dom 𝐹))
16	5, 6, 7, 8, 9, 10, 11	ordtypelem3 9479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
17	15, 16	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (𝐹‘𝑀) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
18	3, 17	sselid 3946	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (𝐹‘𝑀) ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤})
19		breq2 5113	. . . . . . . . . 10 ⊢ (𝑤 = (𝐹‘𝑀) → (𝑗𝑅𝑤 ↔ 𝑗𝑅(𝐹‘𝑀)))
20	19	ralbidv 3157	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑀) → (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀)))
21	20	elrab 3661	. . . . . . . 8 ⊢ ((𝐹‘𝑀) ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ↔ ((𝐹‘𝑀) ∈ 𝐴 ∧ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀)))
22	21	simprbi 496	. . . . . . 7 ⊢ ((𝐹‘𝑀) ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} → ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀))
23	18, 22	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀))
24	5	tfr1a 8364	. . . . . . . . 9 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
25	24	simpli 483	. . . . . . . 8 ⊢ Fun 𝐹
26		funfn 6548	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
27	25, 26	mpbi 230	. . . . . . 7 ⊢ 𝐹 Fn dom 𝐹
28	24	simpri 485	. . . . . . . . 9 ⊢ Lim dom 𝐹
29		limord 6395	. . . . . . . . 9 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
30	28, 29	ax-mp 5	. . . . . . . 8 ⊢ Ord dom 𝐹
31		inss2 4203	. . . . . . . . . 10 ⊢ (𝑇 ∩ dom 𝐹) ⊆ dom 𝐹
32	13, 31	eqsstrdi 3993	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑂 ⊆ dom 𝐹)
33	32	sselda 3948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ dom 𝐹)
34		ordelss 6350	. . . . . . . 8 ⊢ ((Ord dom 𝐹 ∧ 𝑀 ∈ dom 𝐹) → 𝑀 ⊆ dom 𝐹)
35	30, 33, 34	sylancr 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ⊆ dom 𝐹)
36		breq1 5112	. . . . . . . 8 ⊢ (𝑗 = (𝐹‘𝑎) → (𝑗𝑅(𝐹‘𝑀) ↔ (𝐹‘𝑎)𝑅(𝐹‘𝑀)))
37	36	ralima 7213	. . . . . . 7 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑀 ⊆ dom 𝐹) → (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀) ↔ ∀𝑎 ∈ 𝑀 (𝐹‘𝑎)𝑅(𝐹‘𝑀)))
38	27, 35, 37	sylancr 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀) ↔ ∀𝑎 ∈ 𝑀 (𝐹‘𝑎)𝑅(𝐹‘𝑀)))
39	23, 38	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → ∀𝑎 ∈ 𝑀 (𝐹‘𝑎)𝑅(𝐹‘𝑀))
40	39	adantrr 717	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → ∀𝑎 ∈ 𝑀 (𝐹‘𝑎)𝑅(𝐹‘𝑀))
41		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑁 ∈ 𝑀)
42	2, 40, 41	rspcdva 3592	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝐹‘𝑁)𝑅(𝐹‘𝑀))
43	5, 6, 7, 8, 9, 10, 11	ordtypelem1 9477	. . . . . 6 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇))
44	43	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑂 = (𝐹 ↾ 𝑇))
45	44	fveq1d 6862	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑁) = ((𝐹 ↾ 𝑇)‘𝑁))
46	5, 6, 7, 8, 9, 10, 11	ordtypelem2 9478	. . . . . . 7 ⊢ (𝜑 → Ord 𝑇)
47		inss1 4202	. . . . . . . . . 10 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇
48	13, 47	eqsstrdi 3993	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑂 ⊆ 𝑇)
49	48	sselda 3948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ 𝑇)
50	49	adantrr 717	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑀 ∈ 𝑇)
51		ordelss 6350	. . . . . . 7 ⊢ ((Ord 𝑇 ∧ 𝑀 ∈ 𝑇) → 𝑀 ⊆ 𝑇)
52	46, 50, 51	syl2an2r 685	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑀 ⊆ 𝑇)
53	52, 41	sseldd 3949	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑁 ∈ 𝑇)
54	53	fvresd 6880	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → ((𝐹 ↾ 𝑇)‘𝑁) = (𝐹‘𝑁))
55	45, 54	eqtrd 2765	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑁) = (𝐹‘𝑁))
56	44	fveq1d 6862	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑀) = ((𝐹 ↾ 𝑇)‘𝑀))
57	50	fvresd 6880	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → ((𝐹 ↾ 𝑇)‘𝑀) = (𝐹‘𝑀))
58	56, 57	eqtrd 2765	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑀) = (𝐹‘𝑀))
59	42, 55, 58	3brtr4d 5141	. 2 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑁)𝑅(𝑂‘𝑀))
60	59	expr 456	1 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (𝑁 ∈ 𝑀 → (𝑂‘𝑁)𝑅(𝑂‘𝑀)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  {crab 3408  Vcvv 3450   ∩ cin 3915   ⊆ wss 3916   class class class wbr 5109   ↦ cmpt 5190   Se wse 5591   We wwe 5592  dom cdm 5640  ran crn 5641   ↾ cres 5642   “ cima 5643  Ord word 6333  Oncon0 6334  Lim wlim 6335  Fun wfun 6507   Fn wfn 6508  ‘cfv 6513  ℩crio 7345  recscrecs 8341  OrdIsocoi 9468

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 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

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-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-oi 9469

This theorem is referenced by:  ordtypelem8  9484