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Theorem noinfno 27778
Description: The next several theorems deal with a surreal "infimum". This surreal will ultimately be shown to bound 𝐵 above and bound the restriction of any surreal below. We begin by showing that the given expression actually defines a surreal number. (Contributed by Scott Fenton, 8-Aug-2024.)
Hypothesis
Ref Expression
noinfno.1 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
noinfno ((𝐵 No 𝐵𝑉) → 𝑇 No )
Distinct variable groups:   𝑥,𝐵,𝑦,𝑔,𝑣,𝑢   𝑔,𝑉
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑉(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem noinfno
Dummy variables 𝑎 𝑏 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noinfno.1 . 2 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
2 iftrue 4537 . . . . 5 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
32adantr 480 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
4 simprl 771 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → 𝐵 No )
5 simpl 482 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
6 nominmo 27759 . . . . . . . . 9 (𝐵 No → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
76ad2antrl 728 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
8 reu5 3380 . . . . . . . 8 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
95, 7, 8sylanbrc 583 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
10 riotacl 7405 . . . . . . 7 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
119, 10syl 17 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
124, 11sseldd 3996 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No )
13 1oex 8515 . . . . . . 7 1o ∈ V
1413prid1 4767 . . . . . 6 1o ∈ {1o, 2o}
1514noextend 27726 . . . . 5 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) ∈ No )
1612, 15syl 17 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) ∈ No )
173, 16eqeltrd 2839 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) ∈ No )
18 iffalse 4540 . . . . 5 (¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
1918adantr 480 . . . 4 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
20 funmpt 6606 . . . . . 6 Fun (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))
2120a1i 11 . . . . 5 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → Fun (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
22 iotaex 6536 . . . . . . 7 (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ∈ V
23 eqid 2735 . . . . . . 7 (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))
2422, 23dmmpti 6713 . . . . . 6 dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}
25 simpl 482 . . . . . . . . . . . . . 14 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝐵 No )
26 simprl 771 . . . . . . . . . . . . . 14 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑢𝐵)
2725, 26sseldd 3996 . . . . . . . . . . . . 13 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑢 No )
28 nodmon 27710 . . . . . . . . . . . . 13 (𝑢 No → dom 𝑢 ∈ On)
2927, 28syl 17 . . . . . . . . . . . 12 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → dom 𝑢 ∈ On)
30 simprrl 781 . . . . . . . . . . . 12 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑦 ∈ dom 𝑢)
31 onelon 6411 . . . . . . . . . . . 12 ((dom 𝑢 ∈ On ∧ 𝑦 ∈ dom 𝑢) → 𝑦 ∈ On)
3229, 30, 31syl2anc 584 . . . . . . . . . . 11 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑦 ∈ On)
3332rexlimdvaa 3154 . . . . . . . . . 10 (𝐵 No → (∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ On))
3433abssdv 4078 . . . . . . . . 9 (𝐵 No → {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ On)
35 simplr 769 . . . . . . . . . . . . . . . . 17 (((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) → 𝑎𝑏)
36 ssel2 3990 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 No 𝑢𝐵) → 𝑢 No )
3736adantlr 715 . . . . . . . . . . . . . . . . . . 19 (((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) → 𝑢 No )
3837, 28syl 17 . . . . . . . . . . . . . . . . . 18 (((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) → dom 𝑢 ∈ On)
39 ontr1 6432 . . . . . . . . . . . . . . . . . 18 (dom 𝑢 ∈ On → ((𝑎𝑏𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
4038, 39syl 17 . . . . . . . . . . . . . . . . 17 (((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) → ((𝑎𝑏𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
4135, 40mpand 695 . . . . . . . . . . . . . . . 16 (((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) → (𝑏 ∈ dom 𝑢𝑎 ∈ dom 𝑢))
4241adantrd 491 . . . . . . . . . . . . . . 15 (((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → 𝑎 ∈ dom 𝑢))
43 reseq1 5994 . . . . . . . . . . . . . . . . . . 19 ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎))
44 onelon 6411 . . . . . . . . . . . . . . . . . . . . . . . 24 ((dom 𝑢 ∈ On ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
4538, 44sylan 580 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
46 onsucb 7837 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ∈ On ↔ suc 𝑏 ∈ On)
4745, 46sylib 218 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑏 ∈ On)
48 simpllr 776 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → 𝑎𝑏)
49 eloni 6396 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 ∈ On → Ord 𝑏)
5045, 49syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → Ord 𝑏)
51 ordsucelsuc 7842 . . . . . . . . . . . . . . . . . . . . . . . 24 (Ord 𝑏 → (𝑎𝑏 ↔ suc 𝑎 ∈ suc 𝑏))
5250, 51syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → (𝑎𝑏 ↔ suc 𝑎 ∈ suc 𝑏))
5348, 52mpbid 232 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑎 ∈ suc 𝑏)
54 onelss 6428 . . . . . . . . . . . . . . . . . . . . . 22 (suc 𝑏 ∈ On → (suc 𝑎 ∈ suc 𝑏 → suc 𝑎 ⊆ suc 𝑏))
5547, 53, 54sylc 65 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑎 ⊆ suc 𝑏)
5655resabs1d 6028 . . . . . . . . . . . . . . . . . . . 20 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑢 ↾ suc 𝑎))
5755resabs1d 6028 . . . . . . . . . . . . . . . . . . . 20 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))
5856, 57eqeq12d 2751 . . . . . . . . . . . . . . . . . . 19 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → (((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎) ↔ (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
5943, 58imbitrid 244 . . . . . . . . . . . . . . . . . 18 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
6059imim2d 57 . . . . . . . . . . . . . . . . 17 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) → (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
6160ralimdv 3167 . . . . . . . . . . . . . . . 16 ((((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) ∧ 𝑏 ∈ dom 𝑢) → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) → ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
6261expimpd 453 . . . . . . . . . . . . . . 15 (((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
6342, 62jcad 512 . . . . . . . . . . . . . 14 (((𝐵 No 𝑎𝑏) ∧ 𝑢𝐵) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
6463reximdva 3166 . . . . . . . . . . . . 13 ((𝐵 No 𝑎𝑏) → (∃𝑢𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → ∃𝑢𝐵 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
6564expimpd 453 . . . . . . . . . . . 12 (𝐵 No → ((𝑎𝑏 ∧ ∃𝑢𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))) → ∃𝑢𝐵 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
66 vex 3482 . . . . . . . . . . . . . 14 𝑏 ∈ V
67 eleq1w 2822 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏 → (𝑦 ∈ dom 𝑢𝑏 ∈ dom 𝑢))
68 suceq 6452 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
6968reseq2d 6000 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑏))
7068reseq2d 6000 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑏))
7169, 70eqeq12d 2751 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
7271imbi2d 340 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
7372ralbidv 3176 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
7467, 73anbi12d 632 . . . . . . . . . . . . . . 15 (𝑦 = 𝑏 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
7574rexbidv 3177 . . . . . . . . . . . . . 14 (𝑦 = 𝑏 → (∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
7666, 75elab 3681 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
7776anbi2i 623 . . . . . . . . . . . 12 ((𝑎𝑏𝑏 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) ↔ (𝑎𝑏 ∧ ∃𝑢𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
78 vex 3482 . . . . . . . . . . . . 13 𝑎 ∈ V
79 eleq1w 2822 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (𝑦 ∈ dom 𝑢𝑎 ∈ dom 𝑢))
80 suceq 6452 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑎 → suc 𝑦 = suc 𝑎)
8180reseq2d 6000 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑎 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑎))
8280reseq2d 6000 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑎 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑎))
8381, 82eqeq12d 2751 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑎 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
8483imbi2d 340 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑎 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
8584ralbidv 3176 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
8679, 85anbi12d 632 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
8786rexbidv 3177 . . . . . . . . . . . . 13 (𝑦 = 𝑎 → (∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢𝐵 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
8878, 87elab 3681 . . . . . . . . . . . 12 (𝑎 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢𝐵 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
8965, 77, 883imtr4g 296 . . . . . . . . . . 11 (𝐵 No → ((𝑎𝑏𝑏 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
9089alrimivv 1926 . . . . . . . . . 10 (𝐵 No → ∀𝑎𝑏((𝑎𝑏𝑏 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
91 dftr2 5267 . . . . . . . . . 10 (Tr {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∀𝑎𝑏((𝑎𝑏𝑏 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
9290, 91sylibr 234 . . . . . . . . 9 (𝐵 No → Tr {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
93 dford5 7803 . . . . . . . . 9 (Ord {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ({𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ On ∧ Tr {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
9434, 92, 93sylanbrc 583 . . . . . . . 8 (𝐵 No → Ord {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
9594ad2antrl 728 . . . . . . 7 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → Ord {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
96 bdayfo 27737 . . . . . . . . . . . . . 14 bday : No onto→On
97 fofun 6822 . . . . . . . . . . . . . 14 ( bday : No onto→On → Fun bday )
9896, 97ax-mp 5 . . . . . . . . . . . . 13 Fun bday
99 funimaexg 6654 . . . . . . . . . . . . 13 ((Fun bday 𝐵𝑉) → ( bday 𝐵) ∈ V)
10098, 99mpan 690 . . . . . . . . . . . 12 (𝐵𝑉 → ( bday 𝐵) ∈ V)
101100uniexd 7761 . . . . . . . . . . 11 (𝐵𝑉 ( bday 𝐵) ∈ V)
102101adantl 481 . . . . . . . . . 10 ((𝐵 No 𝐵𝑉) → ( bday 𝐵) ∈ V)
103102adantl 481 . . . . . . . . 9 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → ( bday 𝐵) ∈ V)
104 bdayval 27708 . . . . . . . . . . . . . . . 16 (𝑢 No → ( bday 𝑢) = dom 𝑢)
10527, 104syl 17 . . . . . . . . . . . . . . 15 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → ( bday 𝑢) = dom 𝑢)
106 fofn 6823 . . . . . . . . . . . . . . . . . 18 ( bday : No onto→On → bday Fn No )
10796, 106ax-mp 5 . . . . . . . . . . . . . . . . 17 bday Fn No
108 fnfvima 7253 . . . . . . . . . . . . . . . . 17 (( bday Fn No 𝐵 No 𝑢𝐵) → ( bday 𝑢) ∈ ( bday 𝐵))
109107, 108mp3an1 1447 . . . . . . . . . . . . . . . 16 ((𝐵 No 𝑢𝐵) → ( bday 𝑢) ∈ ( bday 𝐵))
110109adantrr 717 . . . . . . . . . . . . . . 15 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → ( bday 𝑢) ∈ ( bday 𝐵))
111105, 110eqeltrrd 2840 . . . . . . . . . . . . . 14 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → dom 𝑢 ∈ ( bday 𝐵))
112 elssuni 4942 . . . . . . . . . . . . . 14 (dom 𝑢 ∈ ( bday 𝐵) → dom 𝑢 ( bday 𝐵))
113111, 112syl 17 . . . . . . . . . . . . 13 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → dom 𝑢 ( bday 𝐵))
114113, 30sseldd 3996 . . . . . . . . . . . 12 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑦 ( bday 𝐵))
115114rexlimdvaa 3154 . . . . . . . . . . 11 (𝐵 No → (∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ( bday 𝐵)))
116115abssdv 4078 . . . . . . . . . 10 (𝐵 No → {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ ( bday 𝐵))
117116ad2antrl 728 . . . . . . . . 9 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ ( bday 𝐵))
118103, 117ssexd 5330 . . . . . . . 8 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ V)
119 elong 6394 . . . . . . . 8 ({𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ V → ({𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On ↔ Ord {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
120118, 119syl 17 . . . . . . 7 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → ({𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On ↔ Ord {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
12195, 120mpbird 257 . . . . . 6 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On)
12224, 121eqeltrid 2843 . . . . 5 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ∈ On)
12323rnmpt 5971 . . . . . 6 ran (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = {𝑧 ∣ ∃𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))}
124 eleq1w 2822 . . . . . . . . . . 11 (𝑦 = 𝑔 → (𝑦 ∈ dom 𝑢𝑔 ∈ dom 𝑢))
125 suceq 6452 . . . . . . . . . . . . . . 15 (𝑦 = 𝑔 → suc 𝑦 = suc 𝑔)
126125reseq2d 6000 . . . . . . . . . . . . . 14 (𝑦 = 𝑔 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑔))
127125reseq2d 6000 . . . . . . . . . . . . . 14 (𝑦 = 𝑔 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑔))
128126, 127eqeq12d 2751 . . . . . . . . . . . . 13 (𝑦 = 𝑔 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))
129128imbi2d 340 . . . . . . . . . . . 12 (𝑦 = 𝑔 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
130129ralbidv 3176 . . . . . . . . . . 11 (𝑦 = 𝑔 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
131124, 130anbi12d 632 . . . . . . . . . 10 (𝑦 = 𝑔 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))))
132131rexbidv 3177 . . . . . . . . 9 (𝑦 = 𝑔 → (∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))))
133132rexab 3703 . . . . . . . 8 (∃𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ↔ ∃𝑔(∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) ∧ 𝑧 = (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
134 eqid 2735 . . . . . . . . . . . . . . . . . 18 (𝑢𝑔) = (𝑢𝑔)
135 fvex 6920 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑔) ∈ V
136 eqeq2 2747 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (𝑢𝑔) → ((𝑢𝑔) = 𝑥 ↔ (𝑢𝑔) = (𝑢𝑔)))
1371363anbi3d 1441 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑢𝑔) → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = (𝑢𝑔))))
138135, 137spcev 3606 . . . . . . . . . . . . . . . . . 18 ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = (𝑢𝑔)) → ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
139134, 138mp3an3 1449 . . . . . . . . . . . . . . . . 17 ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
140139reximi 3082 . . . . . . . . . . . . . . . 16 (∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑢𝐵𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
141 rexcom4 3286 . . . . . . . . . . . . . . . 16 (∃𝑢𝐵𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ ∃𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
142140, 141sylib 218 . . . . . . . . . . . . . . 15 (∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
143142adantl 481 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) ∧ ∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
144 noinfprefixmo 27761 . . . . . . . . . . . . . . . 16 (𝐵 No → ∃*𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
145144ad2antrl 728 . . . . . . . . . . . . . . 15 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → ∃*𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
146145adantr 480 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) ∧ ∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃*𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
147 df-eu 2567 . . . . . . . . . . . . . 14 (∃!𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ (∃𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ∧ ∃*𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))
148143, 146, 147sylanbrc 583 . . . . . . . . . . . . 13 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) ∧ ∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃!𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
149 vex 3482 . . . . . . . . . . . . . 14 𝑧 ∈ V
150 eqeq2 2747 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((𝑢𝑔) = 𝑥 ↔ (𝑢𝑔) = 𝑧))
1511503anbi3d 1441 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧)))
152151rexbidv 3177 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ ∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧)))
153152iota2 6552 . . . . . . . . . . . . . 14 ((𝑧 ∈ V ∧ ∃!𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) → (∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) ↔ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = 𝑧))
154149, 153mpan 690 . . . . . . . . . . . . 13 (∃!𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) → (∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) ↔ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = 𝑧))
155148, 154syl 17 . . . . . . . . . . . 12 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) ∧ ∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) ↔ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = 𝑧))
156 eqcom 2742 . . . . . . . . . . . 12 ((℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = 𝑧𝑧 = (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))
157155, 156bitrdi 287 . . . . . . . . . . 11 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) ∧ ∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) ↔ 𝑧 = (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
158 simprr3 1222 . . . . . . . . . . . . . . 15 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → (𝑢𝑔) = 𝑧)
159 simpl 482 . . . . . . . . . . . . . . . . . 18 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → 𝐵 No )
160 simprl 771 . . . . . . . . . . . . . . . . . 18 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → 𝑢𝐵)
161159, 160sseldd 3996 . . . . . . . . . . . . . . . . 17 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → 𝑢 No )
162 norn 27711 . . . . . . . . . . . . . . . . 17 (𝑢 No → ran 𝑢 ⊆ {1o, 2o})
163161, 162syl 17 . . . . . . . . . . . . . . . 16 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → ran 𝑢 ⊆ {1o, 2o})
164 nofun 27709 . . . . . . . . . . . . . . . . . 18 (𝑢 No → Fun 𝑢)
165161, 164syl 17 . . . . . . . . . . . . . . . . 17 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → Fun 𝑢)
166 simprr1 1220 . . . . . . . . . . . . . . . . 17 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → 𝑔 ∈ dom 𝑢)
167 fvelrn 7096 . . . . . . . . . . . . . . . . 17 ((Fun 𝑢𝑔 ∈ dom 𝑢) → (𝑢𝑔) ∈ ran 𝑢)
168165, 166, 167syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → (𝑢𝑔) ∈ ran 𝑢)
169163, 168sseldd 3996 . . . . . . . . . . . . . . 15 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → (𝑢𝑔) ∈ {1o, 2o})
170158, 169eqeltrrd 2840 . . . . . . . . . . . . . 14 ((𝐵 No ∧ (𝑢𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → 𝑧 ∈ {1o, 2o})
171170rexlimdvaa 3154 . . . . . . . . . . . . 13 (𝐵 No → (∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) → 𝑧 ∈ {1o, 2o}))
172171ad2antrl 728 . . . . . . . . . . . 12 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → (∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) → 𝑧 ∈ {1o, 2o}))
173172adantr 480 . . . . . . . . . . 11 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) ∧ ∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) → 𝑧 ∈ {1o, 2o}))
174157, 173sylbird 260 . . . . . . . . . 10 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) ∧ ∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (𝑧 = (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) → 𝑧 ∈ {1o, 2o}))
175174expimpd 453 . . . . . . . . 9 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → ((∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) ∧ 𝑧 = (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) → 𝑧 ∈ {1o, 2o}))
176175exlimdv 1931 . . . . . . . 8 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → (∃𝑔(∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) ∧ 𝑧 = (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) → 𝑧 ∈ {1o, 2o}))
177133, 176biimtrid 242 . . . . . . 7 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → (∃𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) → 𝑧 ∈ {1o, 2o}))
178177abssdv 4078 . . . . . 6 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → {𝑧 ∣ ∃𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))} ⊆ {1o, 2o})
179123, 178eqsstrid 4044 . . . . 5 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → ran (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ⊆ {1o, 2o})
180 elno2 27714 . . . . 5 ((𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ∈ No ↔ (Fun (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ∧ dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ∈ On ∧ ran (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ⊆ {1o, 2o}))
18121, 122, 179, 180syl3anbrc 1342 . . . 4 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ∈ No )
18219, 181eqeltrd 2839 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉)) → if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) ∈ No )
18317, 182pm2.61ian 812 . 2 ((𝐵 No 𝐵𝑉) → if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) ∈ No )
1841, 183eqeltrid 2843 1 ((𝐵 No 𝐵𝑉) → 𝑇 No )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wex 1776  wcel 2106  ∃*wmo 2536  ∃!weu 2566  {cab 2712  wral 3059  wrex 3068  ∃!wreu 3376  ∃*wrmo 3377  Vcvv 3478  cun 3961  wss 3963  ifcif 4531  {csn 4631  {cpr 4633  cop 4637   cuni 4912   class class class wbr 5148  cmpt 5231  Tr wtr 5265  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Ord word 6385  Oncon0 6386  suc csuc 6388  cio 6514  Fun wfun 6557   Fn wfn 6558  ontowfo 6561  cfv 6563  crio 7387  1oc1o 8498  2oc2o 8499   No csur 27699   <s cslt 27700   bday cbday 27701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-riota 7388  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704
This theorem is referenced by:  noinfbday  27780  noinfres  27782  noinfbnd1lem1  27783  noinfbnd1lem2  27784  noinfbnd1lem3  27785  noinfbnd1lem4  27786  noinfbnd1lem5  27787  noinfbnd1lem6  27788  noinfbnd2  27791  nosupinfsep  27792  noetainflem1  27797  noetainflem2  27798  noetainflem3  27799  noetainflem4  27800  noetalem1  27801
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