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Theorem ordtypelem7 9564
Description: Lemma for ordtype 9572. 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 3961 . . . . . 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 9561 . . . . . . . . . . 11 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
109adantr 480 . . . . . . . . . 10 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
1110fdmd 6746 . . . . . . . . 9 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → dom 𝑂 = (𝑇 ∩ dom 𝐹))
12 inss1 4237 . . . . . . . . . 10 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
132, 3, 4, 5, 6, 7, 8ordtypelem2 9559 . . . . . . . . . . . 12 (𝜑 → Ord 𝑇)
1413adantr 480 . . . . . . . . . . 11 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → Ord 𝑇)
15 ordsson 7803 . . . . . . . . . . 11 (Ord 𝑇𝑇 ⊆ On)
1614, 15syl 17 . . . . . . . . . 10 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → 𝑇 ⊆ On)
1712, 16sstrid 3995 . . . . . . . . 9 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑇 ∩ dom 𝐹) ⊆ On)
1811, 17eqsstrd 4018 . . . . . . . 8 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → dom 𝑂 ⊆ On)
1918sseld 3982 . . . . . . 7 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂𝑀 ∈ On))
20 eleq1 2829 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂))
21 fveq2 6906 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝑂𝑎) = (𝑂𝑏))
2221breq1d 5153 . . . . . . . . . . 11 (𝑎 = 𝑏 → ((𝑂𝑎)𝑅𝑁 ↔ (𝑂𝑏)𝑅𝑁))
2320, 22imbi12d 344 . . . . . . . . . 10 (𝑎 = 𝑏 → ((𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁) ↔ (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)))
2423imbi2d 340 . . . . . . . . 9 (𝑎 = 𝑏 → (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁)) ↔ ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁))))
25 eleq1 2829 . . . . . . . . . . 11 (𝑎 = 𝑀 → (𝑎 ∈ dom 𝑂𝑀 ∈ dom 𝑂))
26 fveq2 6906 . . . . . . . . . . . 12 (𝑎 = 𝑀 → (𝑂𝑎) = (𝑂𝑀))
2726breq1d 5153 . . . . . . . . . . 11 (𝑎 = 𝑀 → ((𝑂𝑎)𝑅𝑁 ↔ (𝑂𝑀)𝑅𝑁))
2825, 27imbi12d 344 . . . . . . . . . 10 (𝑎 = 𝑀 → ((𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁) ↔ (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁)))
2928imbi2d 340 . . . . . . . . 9 (𝑎 = 𝑀 → (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁)) ↔ ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁))))
30 r19.21v 3180 . . . . . . . . . 10 (∀𝑏𝑎 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)) ↔ ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → ∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)))
312tfr1a 8434 . . . . . . . . . . . . . . . . . . . . . . 23 (Fun 𝐹 ∧ Lim dom 𝐹)
3231simpri 485 . . . . . . . . . . . . . . . . . . . . . 22 Lim dom 𝐹
33 limord 6444 . . . . . . . . . . . . . . . . . . . . . 22 (Lim dom 𝐹 → Ord dom 𝐹)
3432, 33ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 Ord dom 𝐹
35 ordin 6414 . . . . . . . . . . . . . . . . . . . . 21 ((Ord 𝑇 ∧ Ord dom 𝐹) → Ord (𝑇 ∩ dom 𝐹))
3614, 34, 35sylancl 586 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → Ord (𝑇 ∩ dom 𝐹))
37 ordeq 6391 . . . . . . . . . . . . . . . . . . . . 21 (dom 𝑂 = (𝑇 ∩ dom 𝐹) → (Ord dom 𝑂 ↔ Ord (𝑇 ∩ dom 𝐹)))
3811, 37syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (Ord dom 𝑂 ↔ Ord (𝑇 ∩ dom 𝐹)))
3936, 38mpbird 257 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → Ord dom 𝑂)
40 ordelss 6400 . . . . . . . . . . . . . . . . . . 19 ((Ord dom 𝑂𝑎 ∈ dom 𝑂) → 𝑎 ⊆ dom 𝑂)
4139, 40sylan 580 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → 𝑎 ⊆ dom 𝑂)
4241sselda 3983 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) ∧ 𝑏𝑎) → 𝑏 ∈ dom 𝑂)
43 pm5.5 361 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ dom 𝑂 → ((𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) ↔ (𝑂𝑏)𝑅𝑁))
4442, 43syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) ∧ 𝑏𝑎) → ((𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) ↔ (𝑂𝑏)𝑅𝑁))
4544ralbidva 3176 . . . . . . . . . . . . . . 15 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → (∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) ↔ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁))
46 eldifn 4132 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (𝐴 ∖ ran 𝑂) → ¬ 𝑁 ∈ ran 𝑂)
4746ad2antlr 727 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ¬ 𝑁 ∈ ran 𝑂)
489ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
4948ffnd 6737 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑂 Fn (𝑇 ∩ dom 𝐹))
50 simprl 771 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ∈ dom 𝑂)
5148fdmd 6746 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → dom 𝑂 = (𝑇 ∩ dom 𝐹))
5250, 51eleqtrd 2843 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹))
53 fnfvelrn 7100 . . . . . . . . . . . . . . . . . . . 20 ((𝑂 Fn (𝑇 ∩ dom 𝐹) ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝑂𝑎) ∈ ran 𝑂)
5449, 52, 53syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) ∈ ran 𝑂)
55 eleq1 2829 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑎) = 𝑁 → ((𝑂𝑎) ∈ ran 𝑂𝑁 ∈ ran 𝑂))
5654, 55syl5ibcom 245 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ((𝑂𝑎) = 𝑁𝑁 ∈ ran 𝑂))
5747, 56mtod 198 . . . . . . . . . . . . . . . . 17 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ¬ (𝑂𝑎) = 𝑁)
58 breq1 5146 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑁 → (𝑢𝑅(𝑂𝑎) ↔ 𝑁𝑅(𝑂𝑎)))
5958notbid 318 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑁 → (¬ 𝑢𝑅(𝑂𝑎) ↔ ¬ 𝑁𝑅(𝑂𝑎)))
602, 3, 4, 5, 6, 7, 8ordtypelem1 9558 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑂 = (𝐹𝑇))
6160ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑂 = (𝐹𝑇))
6261fveq1d 6908 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) = ((𝐹𝑇)‘𝑎))
6352elin1d 4204 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎𝑇)
6463fvresd 6926 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ((𝐹𝑇)‘𝑎) = (𝐹𝑎))
6562, 64eqtrd 2777 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) = (𝐹𝑎))
66 simpll 767 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝜑)
672, 3, 4, 5, 6, 7, 8ordtypelem3 9560 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
6866, 52, 67syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝐹𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
6965, 68eqeltrd 2841 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
70 breq2 5147 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (𝑂𝑎) → (𝑢𝑅𝑣𝑢𝑅(𝑂𝑎)))
7170notbid 318 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = (𝑂𝑎) → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑢𝑅(𝑂𝑎)))
7271ralbidv 3178 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝑂𝑎) → (∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣 ↔ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂𝑎)))
7372elrab 3692 . . . . . . . . . . . . . . . . . . . 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)))
7473simprbi 496 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} → ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂𝑎))
7569, 74syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂𝑎))
76 breq2 5147 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑁 → (𝑗𝑅𝑤𝑗𝑅𝑁))
7776ralbidv 3178 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑁 → (∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁))
78 eldifi 4131 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (𝐴 ∖ ran 𝑂) → 𝑁𝐴)
7978ad2antlr 727 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑁𝐴)
80 simprr 773 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)
8141adantrr 717 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ⊆ dom 𝑂)
8248, 81fssdmd 6754 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ⊆ (𝑇 ∩ dom 𝐹))
8382, 12sstrdi 3996 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎𝑇)
84 fveq1 6905 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑂 = (𝐹𝑇) → (𝑂𝑏) = ((𝐹𝑇)‘𝑏))
85 ssel2 3978 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎𝑇𝑏𝑎) → 𝑏𝑇)
8685fvresd 6926 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎𝑇𝑏𝑎) → ((𝐹𝑇)‘𝑏) = (𝐹𝑏))
8784, 86sylan9eq 2797 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑂 = (𝐹𝑇) ∧ (𝑎𝑇𝑏𝑎)) → (𝑂𝑏) = (𝐹𝑏))
8887anassrs 467 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑂 = (𝐹𝑇) ∧ 𝑎𝑇) ∧ 𝑏𝑎) → (𝑂𝑏) = (𝐹𝑏))
8988breq1d 5153 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑂 = (𝐹𝑇) ∧ 𝑎𝑇) ∧ 𝑏𝑎) → ((𝑂𝑏)𝑅𝑁 ↔ (𝐹𝑏)𝑅𝑁))
9089ralbidva 3176 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑂 = (𝐹𝑇) ∧ 𝑎𝑇) → (∀𝑏𝑎 (𝑂𝑏)𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
9161, 83, 90syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (∀𝑏𝑎 (𝑂𝑏)𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
9280, 91mpbid 232 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁)
9331simpli 483 . . . . . . . . . . . . . . . . . . . . . 22 Fun 𝐹
94 funfn 6596 . . . . . . . . . . . . . . . . . . . . . 22 (Fun 𝐹𝐹 Fn dom 𝐹)
9593, 94mpbi 230 . . . . . . . . . . . . . . . . . . . . 21 𝐹 Fn dom 𝐹
96 inss2 4238 . . . . . . . . . . . . . . . . . . . . . 22 (𝑇 ∩ dom 𝐹) ⊆ dom 𝐹
9782, 96sstrdi 3996 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑎 ⊆ dom 𝐹)
98 breq1 5146 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = (𝐹𝑏) → (𝑗𝑅𝑁 ↔ (𝐹𝑏)𝑅𝑁))
9998ralima 7257 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 Fn dom 𝐹𝑎 ⊆ dom 𝐹) → (∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
10095, 97, 99sylancr 587 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁 ↔ ∀𝑏𝑎 (𝐹𝑏)𝑅𝑁))
10192, 100mpbird 257 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑁)
10277, 79, 101elrabd 3694 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑁 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤})
10359, 75, 102rspcdva 3623 . . . . . . . . . . . . . . . . 17 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ¬ 𝑁𝑅(𝑂𝑎))
104 weso 5676 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 We 𝐴𝑅 Or 𝐴)
1057, 104syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑅 Or 𝐴)
106105ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → 𝑅 Or 𝐴)
10748, 52ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎) ∈ 𝐴)
108 sotric 5622 . . . . . . . . . . . . . . . . . . 19 ((𝑅 Or 𝐴 ∧ ((𝑂𝑎) ∈ 𝐴𝑁𝐴)) → ((𝑂𝑎)𝑅𝑁 ↔ ¬ ((𝑂𝑎) = 𝑁𝑁𝑅(𝑂𝑎))))
109106, 107, 79, 108syl12anc 837 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ((𝑂𝑎)𝑅𝑁 ↔ ¬ ((𝑂𝑎) = 𝑁𝑁𝑅(𝑂𝑎))))
110 ioran 986 . . . . . . . . . . . . . . . . . 18 (¬ ((𝑂𝑎) = 𝑁𝑁𝑅(𝑂𝑎)) ↔ (¬ (𝑂𝑎) = 𝑁 ∧ ¬ 𝑁𝑅(𝑂𝑎)))
111109, 110bitrdi 287 . . . . . . . . . . . . . . . . 17 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → ((𝑂𝑎)𝑅𝑁 ↔ (¬ (𝑂𝑎) = 𝑁 ∧ ¬ 𝑁𝑅(𝑂𝑎))))
11257, 103, 111mpbir2and 713 . . . . . . . . . . . . . . . 16 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏𝑎 (𝑂𝑏)𝑅𝑁)) → (𝑂𝑎)𝑅𝑁)
113112expr 456 . . . . . . . . . . . . . . 15 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → (∀𝑏𝑎 (𝑂𝑏)𝑅𝑁 → (𝑂𝑎)𝑅𝑁))
11445, 113sylbid 240 . . . . . . . . . . . . . 14 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → (∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) → (𝑂𝑎)𝑅𝑁))
115114ex 412 . . . . . . . . . . . . 13 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) → (𝑂𝑎)𝑅𝑁)))
116115com23 86 . . . . . . . . . . . 12 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁)))
117116a2i 14 . . . . . . . . . . 11 (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → ∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)) → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁)))
118117a1i 11 . . . . . . . . . 10 (𝑎 ∈ On → (((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → ∀𝑏𝑎 (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)) → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁))))
11930, 118biimtrid 242 . . . . . . . . 9 (𝑎 ∈ On → (∀𝑏𝑎 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑏 ∈ dom 𝑂 → (𝑂𝑏)𝑅𝑁)) → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂𝑎)𝑅𝑁))))
12024, 29, 119tfis3 7879 . . . . . . . 8 (𝑀 ∈ On → ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁)))
121120com3l 89 . . . . . . 7 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑀 ∈ On → (𝑂𝑀)𝑅𝑁)))
12219, 121mpdd 43 . . . . . 6 ((𝜑𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁))
1231, 122sylan2br 595 . . . . 5 ((𝜑 ∧ (𝑁𝐴 ∧ ¬ 𝑁 ∈ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁))
124123anassrs 467 . . . 4 (((𝜑𝑁𝐴) ∧ ¬ 𝑁 ∈ ran 𝑂) → (𝑀 ∈ dom 𝑂 → (𝑂𝑀)𝑅𝑁))
125124impancom 451 . . 3 (((𝜑𝑁𝐴) ∧ 𝑀 ∈ dom 𝑂) → (¬ 𝑁 ∈ ran 𝑂 → (𝑂𝑀)𝑅𝑁))
126125orrd 864 . 2 (((𝜑𝑁𝐴) ∧ 𝑀 ∈ dom 𝑂) → (𝑁 ∈ ran 𝑂 ∨ (𝑂𝑀)𝑅𝑁))
127126orcomd 872 1 (((𝜑𝑁𝐴) ∧ 𝑀 ∈ dom 𝑂) → ((𝑂𝑀)𝑅𝑁𝑁 ∈ ran 𝑂))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  cin 3950  wss 3951   class class class wbr 5143  cmpt 5225   Or wor 5591   Se wse 5635   We wwe 5636  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Ord word 6383  Oncon0 6384  Lim wlim 6385  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  crio 7387  recscrecs 8410  OrdIsocoi 9549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-oi 9550
This theorem is referenced by:  ordtypelem9  9566  ordtypelem10  9567  oiiniseg  9573
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