Theorem ordtypelem6 9592

Description: Lemma for ordtype 9601. (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 6920	. . . . 5 ⊢ (𝑎 = 𝑁 → (𝐹‘𝑎) = (𝐹‘𝑁))
2	1	breq1d 5176	. . . 4 ⊢ (𝑎 = 𝑁 → ((𝐹‘𝑎)𝑅(𝐹‘𝑀) ↔ (𝐹‘𝑁)𝑅(𝐹‘𝑀)))
3		ssrab2 4103	. . . . . . . 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 9590	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
13	12	fdmd 6757	. . . . . . . . . . 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 2846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ (𝑇 ∩ dom 𝐹))
16	5, 6, 7, 8, 9, 10, 11	ordtypelem3 9589	. . . . . . . . 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 590	. . . . . . . 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 4006	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (𝐹‘𝑀) ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤})
19		breq2 5170	. . . . . . . . . 10 ⊢ (𝑤 = (𝐹‘𝑀) → (𝑗𝑅𝑤 ↔ 𝑗𝑅(𝐹‘𝑀)))
20	19	ralbidv 3184	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑀) → (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀)))
21	20	elrab 3708	. . . . . . . 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 8450	. . . . . . . . 9 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
25	24	simpli 483	. . . . . . . 8 ⊢ Fun 𝐹
26		funfn 6608	. . . . . . . 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 6455	. . . . . . . . 9 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
30	28, 29	ax-mp 5	. . . . . . . 8 ⊢ Ord dom 𝐹
31		inss2 4259	. . . . . . . . . 10 ⊢ (𝑇 ∩ dom 𝐹) ⊆ dom 𝐹
32	13, 31	eqsstrdi 4063	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑂 ⊆ dom 𝐹)
33	32	sselda 4008	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ dom 𝐹)
34		ordelss 6411	. . . . . . . 8 ⊢ ((Ord dom 𝐹 ∧ 𝑀 ∈ dom 𝐹) → 𝑀 ⊆ dom 𝐹)
35	30, 33, 34	sylancr 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ⊆ dom 𝐹)
36		breq1 5169	. . . . . . . 8 ⊢ (𝑗 = (𝐹‘𝑎) → (𝑗𝑅(𝐹‘𝑀) ↔ (𝐹‘𝑎)𝑅(𝐹‘𝑀)))
37	36	ralima 7274	. . . . . . 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 586	. . . . . 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 716	. . . 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 3636	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝐹‘𝑁)𝑅(𝐹‘𝑀))
43	5, 6, 7, 8, 9, 10, 11	ordtypelem1 9587	. . . . . 6 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇))
44	43	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑂 = (𝐹 ↾ 𝑇))
45	44	fveq1d 6922	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑁) = ((𝐹 ↾ 𝑇)‘𝑁))
46	5, 6, 7, 8, 9, 10, 11	ordtypelem2 9588	. . . . . . 7 ⊢ (𝜑 → Ord 𝑇)
47		inss1 4258	. . . . . . . . . 10 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇
48	13, 47	eqsstrdi 4063	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑂 ⊆ 𝑇)
49	48	sselda 4008	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ 𝑇)
50	49	adantrr 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑀 ∈ 𝑇)
51		ordelss 6411	. . . . . . 7 ⊢ ((Ord 𝑇 ∧ 𝑀 ∈ 𝑇) → 𝑀 ⊆ 𝑇)
52	46, 50, 51	syl2an2r 684	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑀 ⊆ 𝑇)
53	52, 41	sseldd 4009	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑁 ∈ 𝑇)
54	53	fvresd 6940	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → ((𝐹 ↾ 𝑇)‘𝑁) = (𝐹‘𝑁))
55	45, 54	eqtrd 2780	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑁) = (𝐹‘𝑁))
56	44	fveq1d 6922	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑀) = ((𝐹 ↾ 𝑇)‘𝑀))
57	50	fvresd 6940	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → ((𝐹 ↾ 𝑇)‘𝑀) = (𝐹‘𝑀))
58	56, 57	eqtrd 2780	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑀) = (𝐹‘𝑀))
59	42, 55, 58	3brtr4d 5198	. 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 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166   ↦ cmpt 5249   Se wse 5650   We wwe 5651  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Ord word 6394  Oncon0 6395  Lim wlim 6396  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573  ℩crio 7403  recscrecs 8426  OrdIsocoi 9578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-oi 9579

This theorem is referenced by:  ordtypelem8  9594