Theorem ordtypelem6 9488

Description: Lemma for ordtype 9497. (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 6885	. . . . 5 ⊢ (𝑎 = 𝑁 → (𝐹‘𝑎) = (𝐹‘𝑁))
2	1	breq1d 5121	. . . 4 ⊢ (𝑎 = 𝑁 → ((𝐹‘𝑎)𝑅(𝐹‘𝑀) ↔ (𝐹‘𝑁)𝑅(𝐹‘𝑀)))
3		ssrab2 4035	. . . . . . . 8 ⊢ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}
4		simpr 490	. . . . . . . . . 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 9486	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
13	12	fdmd 6720	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
14	13	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → dom 𝑂 = (𝑇 ∩ dom 𝐹))
15	4, 14	eleqtrd 2867	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ (𝑇 ∩ dom 𝐹))
16	5, 6, 7, 8, 9, 10, 11	ordtypelem3 9485	. . . . . . . . 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 603	. . . . . . . 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 3936	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (𝐹‘𝑀) ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤})
19		breq2 5115	. . . . . . . . . 10 ⊢ (𝑤 = (𝐹‘𝑀) → (𝑗𝑅𝑤 ↔ 𝑗𝑅(𝐹‘𝑀)))
20	19	ralbidv 3190	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑀) → (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀)))
21	20	elrab 3652	. . . . . . . 8 ⊢ ((𝐹‘𝑀) ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ↔ ((𝐹‘𝑀) ∈ 𝐴 ∧ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀)))
22	21	simprbi 503	. . . . . . 7 ⊢ ((𝐹‘𝑀) ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} → ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀))
23	18, 22	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀))
24	5	tfr1a 8383	. . . . . . . . 9 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
25	24	simpli 489	. . . . . . . 8 ⊢ Fun 𝐹
26		funfn 6570	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
27	25, 26	mpbi 233	. . . . . . 7 ⊢ 𝐹 Fn dom 𝐹
28	24	simpri 491	. . . . . . . . 9 ⊢ Lim dom 𝐹
29		limord 6426	. . . . . . . . 9 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
30	28, 29	ax-mp 5	. . . . . . . 8 ⊢ Ord dom 𝐹
31		inss2 4190	. . . . . . . . . 10 ⊢ (𝑇 ∩ dom 𝐹) ⊆ dom 𝐹
32	13, 31	eqsstrdi 3982	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑂 ⊆ dom 𝐹)
33	32	sselda 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ dom 𝐹)
34		ordelss 6380	. . . . . . . 8 ⊢ ((Ord dom 𝐹 ∧ 𝑀 ∈ dom 𝐹) → 𝑀 ⊆ dom 𝐹)
35	30, 33, 34	sylancr 599	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ⊆ dom 𝐹)
36		breq1 5114	. . . . . . . 8 ⊢ (𝑗 = (𝐹‘𝑎) → (𝑗𝑅(𝐹‘𝑀) ↔ (𝐹‘𝑎)𝑅(𝐹‘𝑀)))
37	36	ralima 7239	. . . . . . 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 599	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀) ↔ ∀𝑎 ∈ 𝑀 (𝐹‘𝑎)𝑅(𝐹‘𝑀)))
39	23, 38	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → ∀𝑎 ∈ 𝑀 (𝐹‘𝑎)𝑅(𝐹‘𝑀))
40	39	adantrr 730	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → ∀𝑎 ∈ 𝑀 (𝐹‘𝑎)𝑅(𝐹‘𝑀))
41		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑁 ∈ 𝑀)
42	2, 40, 41	rspcdva 3584	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝐹‘𝑁)𝑅(𝐹‘𝑀))
43	5, 6, 7, 8, 9, 10, 11	ordtypelem1 9483	. . . . . 6 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇))
44	43	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑂 = (𝐹 ↾ 𝑇))
45	44	fveq1d 6887	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑁) = ((𝐹 ↾ 𝑇)‘𝑁))
46	5, 6, 7, 8, 9, 10, 11	ordtypelem2 9484	. . . . . . 7 ⊢ (𝜑 → Ord 𝑇)
47		inss1 4189	. . . . . . . . . 10 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇
48	13, 47	eqsstrdi 3982	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑂 ⊆ 𝑇)
49	48	sselda 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ 𝑇)
50	49	adantrr 730	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑀 ∈ 𝑇)
51		ordelss 6380	. . . . . . 7 ⊢ ((Ord 𝑇 ∧ 𝑀 ∈ 𝑇) → 𝑀 ⊆ 𝑇)
52	46, 50, 51	syl2an2r 698	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑀 ⊆ 𝑇)
53	52, 41	sseldd 3939	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑁 ∈ 𝑇)
54	53	fvresd 6905	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → ((𝐹 ↾ 𝑇)‘𝑁) = (𝐹‘𝑁))
55	45, 54	eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑁) = (𝐹‘𝑁))
56	44	fveq1d 6887	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑀) = ((𝐹 ↾ 𝑇)‘𝑀))
57	50	fvresd 6905	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → ((𝐹 ↾ 𝑇)‘𝑀) = (𝐹‘𝑀))
58	56, 57	eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑀) = (𝐹‘𝑀))
59	42, 55, 58	3brtr4d 5145	. 2 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑁)𝑅(𝑂‘𝑀))
60	59	expr 462	1 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (𝑁 ∈ 𝑀 → (𝑂‘𝑁)𝑅(𝑂‘𝑀)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5111   ↦ cmpt 5194   Se wse 5614   We wwe 5615  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666  Ord word 6363  Oncon0 6364  Lim wlim 6365  Fun wfun 6534   Fn wfn 6535  ‘cfv 6540  ℩crio 7372  recscrecs 8359  OrdIsocoi 9474

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-oi 9475

This theorem is used by:  ordtypelem8  9490