Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtypelem7 Structured version   Visualization version   GIF version

Theorem ordtypelem7 8596
 Description: Lemma for ordtype 8604. ran 𝑂 is an initial segment of 𝐴 under the well-order 𝑅. (Contributed by Mario Carneiro, 25-Jun-2015.)
Hypotheses
Ref Expression
ordtypelem.1 𝐹 = recs(𝐺)
ordtypelem.2 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
ordtypelem.3 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
ordtypelem.5 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
ordtypelem.6 𝑂 = OrdIso(𝑅, 𝐴)
ordtypelem.7 (𝜑𝑅 We 𝐴)
ordtypelem.8 (𝜑𝑅 Se 𝐴)
Assertion
Ref Expression
ordtypelem7 (((𝜑𝑁𝐴) ∧ 𝑀 ∈ dom 𝑂) → ((𝑂𝑀)𝑅𝑁𝑁 ∈ ran 𝑂))
Distinct variable groups:   𝑣,𝑢,𝐶   ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑀   𝑗,𝑁,𝑢,𝑤   𝑅,,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   𝐴,,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   𝑡,𝑂,𝑢,𝑣,𝑥   𝜑,𝑡,𝑥   ,𝐹,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤,𝑣,𝑢,,𝑗)   𝐶(𝑥,𝑧,𝑤,𝑡,,𝑗)   𝑇(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,,𝑗)   𝐺(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,,𝑗)   𝑁(𝑥,𝑧,𝑣,𝑡,)   𝑂(𝑧,𝑤,,𝑗)

Proof of Theorem ordtypelem7
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldif 3725 . . . . . 6 (𝑁 ∈ (𝐴 ∖ ran 𝑂) ↔ (𝑁𝐴 ∧ ¬ 𝑁 ∈ ran 𝑂))
2 ordtypelem.1 . . . . . . . . . . . 12 𝐹 = recs(𝐺)
3 ordtypelem.2 . . . . . . . . . . . 12 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
4 ordtypelem.3 . . . . . . . . . . . 12 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
5 ordtypelem.5 . . . . . . . . . . . 12 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
6 ordtypelem.6 . . . . . . . . . . . 12 𝑂 = OrdIso(𝑅, 𝐴)
7 ordtypelem.7 . . . . . . . . . . . 12 (𝜑𝑅 We 𝐴)
8 ordtypelem.8 . . . . . . . . . . . 12 (𝜑𝑅 Se 𝐴)
92, 3, 4, 5, 6, 7, 8ordtypelem4 8593 . . . . . . . . . . 11 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
109adantr 472 . . . . . . . . . 10 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
11 fdm 6212 . . . . . . . . . 10 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
1210, 11syl 17 . . . . . . . . 9 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → dom 𝑂 = (𝑇 ∩ dom 𝐹))
13 inss1 3976 . . . . . . . . . 10 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
142, 3, 4, 5, 6, 7, 8ordtypelem2 8591 . . . . . . . . . . . 12 (𝜑 → Ord 𝑇)
1514adantr 472 . . . . . . . . . . 11 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → Ord 𝑇)
16 ordsson 7155 . . . . . . . . . . 11 (Ord 𝑇𝑇 ⊆ On)
1715, 16syl 17 . . . . . . . . . 10 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → 𝑇 ⊆ On)
1813, 17syl5ss 3755 . . . . . . . . 9 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑇 ∩ dom 𝐹) ⊆ On)
1912, 18eqsstrd 3780 . . . . . . . 8 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → dom 𝑂 ⊆ On)
2019sseld 3743 . . . . . . 7 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂𝑀 ∈ On))
21 eleq1 2827 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂))
22 fveq2 6353 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝑂𝑎) = (𝑂𝑏))
2322breq1d 4814 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((𝑂𝑎)𝑅𝑁 ↔ (𝑂𝑏)𝑅𝑁))
2421, 23imbi12d 333 . . . . . . . . . 10 (𝑎 = 𝑏 → ((𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁) ↔ (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)))
2524imbi2d 329 . . . . . . . . 9 (𝑎 = 𝑏 → (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁)) ↔ ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁))))
26 eleq1 2827 . . . . . . . . . . 11 (𝑎 = 𝑀 → (𝑎 ∈ dom 𝑂𝑀 ∈ dom 𝑂))
27 fveq2 6353 . . . . . . . . . . . 12 (𝑎 = 𝑀 → (𝑂𝑎) = (𝑂𝑀))
2827breq1d 4814 . . . . . . . . . . 11 (𝑎 = 𝑀 → ((𝑂𝑎)𝑅𝑁 ↔ (𝑂𝑀)𝑅𝑁))
2926, 28imbi12d 333 . . . . . . . . . 10 (𝑎 = 𝑀 → ((𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁) ↔ (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁)))
3029imbi2d 329 . . . . . . . . 9 (𝑎 = 𝑀 → (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁)) ↔ ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁))))
31 r19.21v 3098 . . . . . . . . . 10 (∀𝑏𝑎 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)) ↔ ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → ∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)))
322tfr1a 7660 . . . . . . . . . . . . . . . . . . . . . . 23 (Fun 𝐹 ∧ Lim dom 𝐹)
3332simpri 481 . . . . . . . . . . . . . . . . . . . . . 22 Lim dom 𝐹
34 limord 5945 . . . . . . . . . . . . . . . . . . . . . 22 (Lim dom 𝐹 → Ord dom 𝐹)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 Ord dom 𝐹
36 ordin 5914 . . . . . . . . . . . . . . . . . . . . 21 ((Ord 𝑇 ∧ Ord dom 𝐹) → Ord (𝑇 ∩ dom 𝐹))
3715, 35, 36sylancl 697 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → Ord (𝑇 ∩ dom 𝐹))
38 ordeq 5891 . . . . . . . . . . . . . . . . . . . . 21 (dom 𝑂 = (𝑇 ∩ dom 𝐹) → (Ord dom 𝑂 ↔ Ord (𝑇 ∩ dom 𝐹)))
3912, 38syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (Ord dom 𝑂 ↔ Ord (𝑇 ∩ dom 𝐹)))
4037, 39mpbird 247 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → Ord dom 𝑂)
41 ordelss 5900 . . . . . . . . . . . . . . . . . . 19 ((Ord dom 𝑂𝑎 ∈ dom 𝑂) → 𝑎 ⊆ dom 𝑂)
4240, 41sylan 489 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → 𝑎 ⊆ dom 𝑂)
4342sselda 3744 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) ∧ 𝑏𝑎) → 𝑏 ∈ dom 𝑂)
44 pm5.5 350 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ dom 𝑂 → ((𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) ↔ (𝑂𝑏)𝑅𝑁))
4543, 44syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) ∧ 𝑏𝑎) → ((𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) ↔ (𝑂𝑏)𝑅𝑁))
4645ralbidva 3123 . . . . . . . . . . . . . . 15 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → (∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) ↔ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁))
47 eldifn 3876 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (𝐴 ∖ ran 𝑂) → ¬ 𝑁 ∈ ran 𝑂)
4847ad2antlr 765 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ¬ 𝑁 ∈ ran 𝑂)
499ad2antrr 764 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
50 ffn 6206 . . . . . . . . . . . . . . . . . . . . 21 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴𝑂 Fn (𝑇 ∩ dom 𝐹))
5149, 50syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑂 Fn (𝑇 ∩ dom 𝐹))
52 simprl 811 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ∈ dom 𝑂)
5349, 11syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → dom 𝑂 = (𝑇 ∩ dom 𝐹))
5452, 53eleqtrd 2841 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹))
55 fnfvelrn 6520 . . . . . . . . . . . . . . . . . . . 20 ((𝑂 Fn (𝑇 ∩ dom 𝐹) ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝑂𝑎) ∈ ran 𝑂)
5651, 54, 55syl2anc 696 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) ∈ ran 𝑂)
57 eleq1 2827 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑎) = 𝑁 → ((𝑂𝑎) ∈ ran 𝑂𝑁 ∈ ran 𝑂))
5856, 57syl5ibcom 235 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ((𝑂𝑎) = 𝑁𝑁 ∈ ran 𝑂))
5948, 58mtod 189 . . . . . . . . . . . . . . . . 17 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ¬ (𝑂𝑎) = 𝑁)
60 breq1 4807 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑁 → (𝑢𝑅(𝑂𝑎) ↔ 𝑁𝑅(𝑂𝑎)))
6160notbid 307 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑁 → (¬ 𝑢𝑅(𝑂𝑎) ↔ ¬ 𝑁𝑅(𝑂𝑎)))
622, 3, 4, 5, 6, 7, 8ordtypelem1 8590 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑂 = (𝐹𝑇))
6362ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑂 = (𝐹𝑇))
6463fveq1d 6355 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) = ((𝐹𝑇)‘𝑎))
6513, 54sseldi 3742 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎𝑇)
66 fvres 6369 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎𝑇 → ((𝐹𝑇)‘𝑎) = (𝐹𝑎))
6765, 66syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ((𝐹𝑇)‘𝑎) = (𝐹𝑎))
6864, 67eqtrd 2794 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) = (𝐹𝑎))
69 simpll 807 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝜑)
702, 3, 4, 5, 6, 7, 8ordtypelem3 8592 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
7169, 54, 70syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝐹𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
7268, 71eqeltrd 2839 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
73 breq2 4808 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (𝑂𝑎) → (𝑢𝑅𝑣𝑢𝑅(𝑂𝑎)))
7473notbid 307 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = (𝑂𝑎) → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑢𝑅(𝑂𝑎)))
7574ralbidv 3124 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝑂𝑎) → (∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣 ↔ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂𝑎)))
7675elrab 3504 . . . . . . . . . . . . . . . . . . . 20 ((𝑂𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ↔ ((𝑂𝑎) ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∧ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂𝑎)))
7776simprbi 483 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} → ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂𝑎))
7872, 77syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂𝑎))
79 eldifi 3875 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (𝐴 ∖ ran 𝑂) → 𝑁𝐴)
8079ad2antlr 765 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑁𝐴)
81 simprr 813 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)
8242adantrr 755 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ⊆ dom 𝑂)
8382, 53sseqtrd 3782 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ⊆ (𝑇 ∩ dom 𝐹))
8483, 13syl6ss 3756 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎𝑇)
85 fveq1 6352 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑂 = (𝐹𝑇) → (𝑂𝑏) = ((𝐹𝑇)‘𝑏))
86 ssel2 3739 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎𝑇𝑏𝑎) → 𝑏𝑇)
87 fvres 6369 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑏𝑇 → ((𝐹𝑇)‘𝑏) = (𝐹𝑏))
8886, 87syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎𝑇𝑏𝑎) → ((𝐹𝑇)‘𝑏) = (𝐹𝑏))
8985, 88sylan9eq 2814 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑂 = (𝐹𝑇) ∧ (𝑎𝑇𝑏𝑎)) → (𝑂𝑏) = (𝐹𝑏))
9089anassrs 683 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑂 = (𝐹𝑇) ∧ 𝑎𝑇) ∧ 𝑏𝑎) → (𝑂𝑏) = (𝐹𝑏))
9190breq1d 4814 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑂 = (𝐹𝑇) ∧ 𝑎𝑇) ∧ 𝑏𝑎) → ((𝑂𝑏)𝑅𝑁 ↔ (𝐹𝑏)𝑅𝑁))
9291ralbidva 3123 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑂 = (𝐹𝑇) ∧ 𝑎𝑇) → (∀𝑏𝑎 (𝑂𝑏)𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
9363, 84, 92syl2anc 696 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (∀𝑏𝑎 (𝑂𝑏)𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
9481, 93mpbid 222 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁)
9532simpli 476 . . . . . . . . . . . . . . . . . . . . . 22 Fun 𝐹
96 funfn 6079 . . . . . . . . . . . . . . . . . . . . . 22 (Fun 𝐹𝐹 Fn dom 𝐹)
9795, 96mpbi 220 . . . . . . . . . . . . . . . . . . . . 21 𝐹 Fn dom 𝐹
98 inss2 3977 . . . . . . . . . . . . . . . . . . . . . 22 (𝑇 ∩ dom 𝐹) ⊆ dom 𝐹
9983, 98syl6ss 3756 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ⊆ dom 𝐹)
100 breq1 4807 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = (𝐹𝑏) → (𝑗𝑅𝑁 ↔ (𝐹𝑏)𝑅𝑁))
101100ralima 6662 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 Fn dom 𝐹𝑎 ⊆ dom 𝐹) → (∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
10297, 99, 101sylancr 698 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
10394, 102mpbird 247 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁)
104 breq2 4808 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑁 → (𝑗𝑅𝑤𝑗𝑅𝑁))
105104ralbidv 3124 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑁 → (∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁))
106105elrab 3504 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ↔ (𝑁𝐴 ∧ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁))
10780, 103, 106sylanbrc 701 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑁 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤})
10861, 78, 107rspcdva 3455 . . . . . . . . . . . . . . . . 17 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ¬ 𝑁𝑅(𝑂𝑎))
109 weso 5257 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 We 𝐴𝑅 Or 𝐴)
1107, 109syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑅 Or 𝐴)
111110ad2antrr 764 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑅 Or 𝐴)
11249, 54ffvelrnd 6524 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) ∈ 𝐴)
113 sotric 5213 . . . . . . . . . . . . . . . . . . 19 ((𝑅 Or 𝐴 ∧ ((𝑂𝑎) ∈ 𝐴𝑁𝐴)) → ((𝑂𝑎)𝑅𝑁 ↔ ¬ ((𝑂𝑎) = 𝑁𝑁𝑅(𝑂𝑎))))
114111, 112, 80, 113syl12anc 1475 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ((𝑂𝑎)𝑅𝑁 ↔ ¬ ((𝑂𝑎) = 𝑁𝑁𝑅(𝑂𝑎))))
115 ioran 512 . . . . . . . . . . . . . . . . . 18 (¬ ((𝑂𝑎) = 𝑁𝑁𝑅(𝑂𝑎)) ↔ (¬ (𝑂𝑎) = 𝑁 ∧ ¬ 𝑁𝑅(𝑂𝑎)))
116114, 115syl6bb 276 . . . . . . . . . . . . . . . . 17 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ((𝑂𝑎)𝑅𝑁 ↔ (¬ (𝑂𝑎) = 𝑁 ∧ ¬ 𝑁𝑅(𝑂𝑎))))
11759, 108, 116mpbir2and 995 . . . . . . . . . . . . . . . 16 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎)𝑅𝑁)
118117expr 644 . . . . . . . . . . . . . . 15 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → (∀𝑏𝑎 (𝑂𝑏)𝑅𝑁 → (𝑂𝑎)𝑅𝑁))
11946, 118sylbid 230 . . . . . . . . . . . . . 14 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → (∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) → (𝑂𝑎)𝑅𝑁))
120119ex 449 . . . . . . . . . . . . 13 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) → (𝑂𝑎)𝑅𝑁)))
121120com23 86 . . . . . . . . . . . 12 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁)))
122121a2i 14 . . . . . . . . . . 11 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → ∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)) → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁)))
123122a1i 11 . . . . . . . . . 10 (𝑎 ∈ On → (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → ∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)) → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁))))
12431, 123syl5bi 232 . . . . . . . . 9 (𝑎 ∈ On → (∀𝑏𝑎 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)) → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁))))
12525, 30, 124tfis3 7223 . . . . . . . 8 (𝑀 ∈ On → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁)))
126125com3l 89 . . . . . . 7 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑀 ∈ On → (𝑂𝑀)𝑅𝑁)))
12720, 126mpdd 43 . . . . . 6 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁))
1281, 127sylan2br 494 . . . . 5 ((𝜑 ∧ (𝑁𝐴 ∧ ¬ 𝑁 ∈ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁))
129128anassrs 683 . . . 4 (((𝜑𝑁𝐴) ∧ ¬ 𝑁 ∈ ran 𝑂) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁))
130129impancom 455 . . 3 (((𝜑𝑁𝐴) ∧ 𝑀 ∈ dom 𝑂) → (¬ 𝑁 ∈ ran 𝑂 → (𝑂𝑀)𝑅𝑁))
131130orrd 392 . 2 (((𝜑𝑁𝐴) ∧ 𝑀 ∈ dom 𝑂) → (𝑁 ∈ ran 𝑂 ∨ (𝑂𝑀)𝑅𝑁))
132131orcomd 402 1 (((𝜑𝑁𝐴) ∧ 𝑀 ∈ dom 𝑂) → ((𝑂𝑀)𝑅𝑁𝑁 ∈ ran 𝑂))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051  {crab 3054  Vcvv 3340   ∖ cdif 3712   ∩ cin 3714   ⊆ wss 3715   class class class wbr 4804   ↦ cmpt 4881   Or wor 5186   Se wse 5223   We wwe 5224  dom cdm 5266  ran crn 5267   ↾ cres 5268   “ cima 5269  Ord word 5883  Oncon0 5884  Lim wlim 5885  Fun wfun 6043   Fn wfn 6044  ⟶wf 6045  ‘cfv 6049  ℩crio 6774  recscrecs 7637  OrdIsocoi 8581 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-wrecs 7577  df-recs 7638  df-oi 8582 This theorem is referenced by:  ordtypelem9  8598  ordtypelem10  8599  oiiniseg  8605
 Copyright terms: Public domain W3C validator