Theorem ordtypelem6 9481

Description: Lemma for ordtype 9490. (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 6881	. . . . 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 4034	. . . . . . . 8 ⊢ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}
4		simpr 489	. . . . . . . . . 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 9479	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
13	12	fdmd 6716	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
14	13	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → dom 𝑂 = (𝑇 ∩ dom 𝐹))
15	4, 14	eleqtrd 2865	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ (𝑇 ∩ dom 𝐹))
16	5, 6, 7, 8, 9, 10, 11	ordtypelem3 9478	. . . . . . . . 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 602	. . . . . . . 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 3935	. . . . . . 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 3188	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑀) → (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀)))
21	20	elrab 3650	. . . . . . . 8 ⊢ ((𝐹‘𝑀) ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ↔ ((𝐹‘𝑀) ∈ 𝐴 ∧ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅(𝐹‘𝑀)))
22	21	simprbi 502	. . . . . . 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 8377	. . . . . . . . 9 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
25	24	simpli 488	. . . . . . . 8 ⊢ Fun 𝐹
26		funfn 6566	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
27	25, 26	mpbi 233	. . . . . . 7 ⊢ 𝐹 Fn dom 𝐹
28	24	simpri 490	. . . . . . . . 9 ⊢ Lim dom 𝐹
29		limord 6422	. . . . . . . . 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 3981	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑂 ⊆ dom 𝐹)
33	32	sselda 3937	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ dom 𝐹)
34		ordelss 6376	. . . . . . . 8 ⊢ ((Ord dom 𝐹 ∧ 𝑀 ∈ dom 𝐹) → 𝑀 ⊆ dom 𝐹)
35	30, 33, 34	sylancr 598	. . . . . . 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 7235	. . . . . . 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 598	. . . . . 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 729	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → ∀𝑎 ∈ 𝑀 (𝐹‘𝑎)𝑅(𝐹‘𝑀))
41		simprr 784	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑁 ∈ 𝑀)
42	2, 40, 41	rspcdva 3582	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝐹‘𝑁)𝑅(𝐹‘𝑀))
43	5, 6, 7, 8, 9, 10, 11	ordtypelem1 9476	. . . . . 6 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇))
44	43	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑂 = (𝐹 ↾ 𝑇))
45	44	fveq1d 6883	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑁) = ((𝐹 ↾ 𝑇)‘𝑁))
46	5, 6, 7, 8, 9, 10, 11	ordtypelem2 9477	. . . . . . 7 ⊢ (𝜑 → Ord 𝑇)
47		inss1 4189	. . . . . . . . . 10 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇
48	13, 47	eqsstrdi 3981	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑂 ⊆ 𝑇)
49	48	sselda 3937	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → 𝑀 ∈ 𝑇)
50	49	adantrr 729	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑀 ∈ 𝑇)
51		ordelss 6376	. . . . . . 7 ⊢ ((Ord 𝑇 ∧ 𝑀 ∈ 𝑇) → 𝑀 ⊆ 𝑇)
52	46, 50, 51	syl2an2r 697	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑀 ⊆ 𝑇)
53	52, 41	sseldd 3938	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → 𝑁 ∈ 𝑇)
54	53	fvresd 6901	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → ((𝐹 ↾ 𝑇)‘𝑁) = (𝐹‘𝑁))
55	45, 54	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑁) = (𝐹‘𝑁))
56	44	fveq1d 6883	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑀) = ((𝐹 ↾ 𝑇)‘𝑀))
57	50	fvresd 6901	. . . 4 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → ((𝐹 ↾ 𝑇)‘𝑀) = (𝐹‘𝑀))
58	56, 57	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑀) = (𝐹‘𝑀))
59	42, 55, 58	3brtr4d 5143	. 2 ⊢ ((𝜑 ∧ (𝑀 ∈ dom 𝑂 ∧ 𝑁 ∈ 𝑀)) → (𝑂‘𝑁)𝑅(𝑂‘𝑀))
60	59	expr 461	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 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905   class class class wbr 5109   ↦ cmpt 5192   Se wse 5612   We wwe 5613  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664  Ord word 6359  Oncon0 6360  Lim wlim 6361  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  ℩crio 7366  recscrecs 8353  OrdIsocoi 9467

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-oi 9468

This theorem is used by:  ordtypelem8  9483