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Theorem ordtypelem7 8594
Description: Lemma for ordtype 8602. 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 8591 . . . . . . . . . . 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 8589 . . . . . . . . . . . 12 (𝜑 → Ord 𝑇)
1514adantr 472 . . . . . . . . . . 11 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → Ord 𝑇)
16 ordsson 7154 . . . . . . . . . . 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 6352 . . . . . . . . . . . 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 6352 . . . . . . . . . . . 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 7659 . . . . . . . . . . . . . . . . . . . . . . 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 6519 . . . . . . . . . . . . . . . . . . . 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 eldifi 3875 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (𝐴 ∖ ran 𝑂) → 𝑁𝐴)
6160ad2antlr 765 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑁𝐴)
62 simprr 813 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)
632, 3, 4, 5, 6, 7, 8ordtypelem1 8588 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑂 = (𝐹𝑇))
6463ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑂 = (𝐹𝑇))
6542adantrr 755 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ⊆ dom 𝑂)
6665, 53sseqtrd 3782 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ⊆ (𝑇 ∩ dom 𝐹))
6766, 13syl6ss 3756 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎𝑇)
68 fveq1 6351 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑂 = (𝐹𝑇) → (𝑂𝑏) = ((𝐹𝑇)‘𝑏))
69 ssel2 3739 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎𝑇𝑏𝑎) → 𝑏𝑇)
70 fvres 6368 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑏𝑇 → ((𝐹𝑇)‘𝑏) = (𝐹𝑏))
7169, 70syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎𝑇𝑏𝑎) → ((𝐹𝑇)‘𝑏) = (𝐹𝑏))
7268, 71sylan9eq 2814 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑂 = (𝐹𝑇) ∧ (𝑎𝑇𝑏𝑎)) → (𝑂𝑏) = (𝐹𝑏))
7372anassrs 683 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑂 = (𝐹𝑇) ∧ 𝑎𝑇) ∧ 𝑏𝑎) → (𝑂𝑏) = (𝐹𝑏))
7473breq1d 4814 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑂 = (𝐹𝑇) ∧ 𝑎𝑇) ∧ 𝑏𝑎) → ((𝑂𝑏)𝑅𝑁 ↔ (𝐹𝑏)𝑅𝑁))
7574ralbidva 3123 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑂 = (𝐹𝑇) ∧ 𝑎𝑇) → (∀𝑏𝑎 (𝑂𝑏)𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
7664, 67, 75syl2anc 696 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (∀𝑏𝑎 (𝑂𝑏)𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
7762, 76mpbid 222 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁)
7832simpli 476 . . . . . . . . . . . . . . . . . . . . . 22 Fun 𝐹
79 funfn 6079 . . . . . . . . . . . . . . . . . . . . . 22 (Fun 𝐹𝐹 Fn dom 𝐹)
8078, 79mpbi 220 . . . . . . . . . . . . . . . . . . . . 21 𝐹 Fn dom 𝐹
81 inss2 3977 . . . . . . . . . . . . . . . . . . . . . 22 (𝑇 ∩ dom 𝐹) ⊆ dom 𝐹
8266, 81syl6ss 3756 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ⊆ dom 𝐹)
83 breq1 4807 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = (𝐹𝑏) → (𝑗𝑅𝑁 ↔ (𝐹𝑏)𝑅𝑁))
8483ralima 6661 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 Fn dom 𝐹𝑎 ⊆ dom 𝐹) → (∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
8580, 82, 84sylancr 698 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
8677, 85mpbird 247 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁)
87 breq2 4808 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑁 → (𝑗𝑅𝑤𝑗𝑅𝑁))
8887ralbidv 3124 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑁 → (∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁))
8988elrab 3504 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ↔ (𝑁𝐴 ∧ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁))
9061, 86, 89sylanbrc 701 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑁 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤})
9164fveq1d 6354 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) = ((𝐹𝑇)‘𝑎))
9213, 54sseldi 3742 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎𝑇)
93 fvres 6368 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎𝑇 → ((𝐹𝑇)‘𝑎) = (𝐹𝑎))
9492, 93syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ((𝐹𝑇)‘𝑎) = (𝐹𝑎))
9591, 94eqtrd 2794 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) = (𝐹𝑎))
96 simpll 807 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝜑)
972, 3, 4, 5, 6, 7, 8ordtypelem3 8590 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
9896, 54, 97syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝐹𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
9995, 98eqeltrd 2839 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
100 breq2 4808 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (𝑂𝑎) → (𝑢𝑅𝑣𝑢𝑅(𝑂𝑎)))
101100notbid 307 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = (𝑂𝑎) → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑢𝑅(𝑂𝑎)))
102101ralbidv 3124 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝑂𝑎) → (∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣 ↔ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂𝑎)))
103102elrab 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)))
104103simprbi 483 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} → ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂𝑎))
10599, 104syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂𝑎))
106 breq1 4807 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑁 → (𝑢𝑅(𝑂𝑎) ↔ 𝑁𝑅(𝑂𝑎)))
107106notbid 307 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑁 → (¬ 𝑢𝑅(𝑂𝑎) ↔ ¬ 𝑁𝑅(𝑂𝑎)))
108107rspcv 3445 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} → (∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂𝑎) → ¬ 𝑁𝑅(𝑂𝑎)))
10990, 105, 108sylc 65 . . . . . . . . . . . . . . . . 17 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ¬ 𝑁𝑅(𝑂𝑎))
110 weso 5257 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 We 𝐴𝑅 Or 𝐴)
1117, 110syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑅 Or 𝐴)
112111ad2antrr 764 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑅 Or 𝐴)
11349, 54ffvelrnd 6523 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) ∈ 𝐴)
114 sotric 5213 . . . . . . . . . . . . . . . . . . 19 ((𝑅 Or 𝐴 ∧ ((𝑂𝑎) ∈ 𝐴𝑁𝐴)) → ((𝑂𝑎)𝑅𝑁 ↔ ¬ ((𝑂𝑎) = 𝑁𝑁𝑅(𝑂𝑎))))
115112, 113, 61, 114syl12anc 1475 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ((𝑂𝑎)𝑅𝑁 ↔ ¬ ((𝑂𝑎) = 𝑁𝑁𝑅(𝑂𝑎))))
116 ioran 512 . . . . . . . . . . . . . . . . . 18 (¬ ((𝑂𝑎) = 𝑁𝑁𝑅(𝑂𝑎)) ↔ (¬ (𝑂𝑎) = 𝑁 ∧ ¬ 𝑁𝑅(𝑂𝑎)))
117115, 116syl6bb 276 . . . . . . . . . . . . . . . . 17 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ((𝑂𝑎)𝑅𝑁 ↔ (¬ (𝑂𝑎) = 𝑁 ∧ ¬ 𝑁𝑅(𝑂𝑎))))
11859, 109, 117mpbir2and 995 . . . . . . . . . . . . . . . 16 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎)𝑅𝑁)
119118expr 644 . . . . . . . . . . . . . . 15 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → (∀𝑏𝑎 (𝑂𝑏)𝑅𝑁 → (𝑂𝑎)𝑅𝑁))
12046, 119sylbid 230 . . . . . . . . . . . . . 14 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → (∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) → (𝑂𝑎)𝑅𝑁))
121120ex 449 . . . . . . . . . . . . 13 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) → (𝑂𝑎)𝑅𝑁)))
122121com23 86 . . . . . . . . . . . 12 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁)))
123122a2i 14 . . . . . . . . . . 11 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → ∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)) → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁)))
124123a1i 11 . . . . . . . . . 10 (𝑎 ∈ On → (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → ∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)) → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁))))
12531, 124syl5bi 232 . . . . . . . . 9 (𝑎 ∈ On → (∀𝑏𝑎 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)) → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁))))
12625, 30, 125tfis3 7222 . . . . . . . 8 (𝑀 ∈ On → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁)))
127126com3l 89 . . . . . . 7 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑀 ∈ On → (𝑂𝑀)𝑅𝑁)))
12820, 127mpdd 43 . . . . . 6 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁))
1291, 128sylan2br 494 . . . . 5 ((𝜑 ∧ (𝑁𝐴 ∧ ¬ 𝑁 ∈ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁))
130129anassrs 683 . . . 4 (((𝜑𝑁𝐴) ∧ ¬ 𝑁 ∈ ran 𝑂) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁))
131130impancom 455 . . 3 (((𝜑𝑁𝐴) ∧ 𝑀 ∈ dom 𝑂) → (¬ 𝑁 ∈ ran 𝑂 → (𝑂𝑀)𝑅𝑁))
132131orrd 392 . 2 (((𝜑𝑁𝐴) ∧ 𝑀 ∈ dom 𝑂) → (𝑁 ∈ ran 𝑂 ∨ (𝑂𝑀)𝑅𝑁))
133132orcomd 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 6773  recscrecs 7636  OrdIsocoi 8579
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 7114
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 6774  df-wrecs 7576  df-recs 7637  df-oi 8580
This theorem is referenced by:  ordtypelem9  8596  ordtypelem10  8597  oiiniseg  8603
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