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Theorem nosupno 33316
Description: The next several theorems deal with a surreal "supremum". This surreal will ultimately be shown to bound 𝐴 below and bound the restriction of any surreal above. We begin by showing that the given expression actually defines a surreal number. (Contributed by Scott Fenton, 5-Dec-2021.)
Hypothesis
Ref Expression
nosupno.1 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
nosupno ((𝐴 No 𝐴𝑉) → 𝑆 No )
Distinct variable group:   𝑥,𝐴,𝑦,𝑔,𝑣,𝑢
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑉(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem nosupno
Dummy variables 𝑎 𝑏 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3459 . 2 (𝐴𝑉𝐴 ∈ V)
2 nosupno.1 . . 3 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
3 iftrue 4431 . . . . . 6 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
43adantr 484 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
5 simprl 770 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → 𝐴 No )
6 simpl 486 . . . . . . . . 9 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
7 nomaxmo 33314 . . . . . . . . . 10 (𝐴 No → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
87ad2antrl 727 . . . . . . . . 9 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
9 reu5 3375 . . . . . . . . 9 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ↔ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
106, 8, 9sylanbrc 586 . . . . . . . 8 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
11 riotacl 7110 . . . . . . . 8 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
1210, 11syl 17 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
135, 12sseldd 3916 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No )
14 2on 8094 . . . . . . . . 9 2o ∈ On
1514elexi 3460 . . . . . . . 8 2o ∈ V
1615prid2 4659 . . . . . . 7 2o ∈ {1o, 2o}
1716noextend 33286 . . . . . 6 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) ∈ No )
1813, 17syl 17 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) ∈ No )
194, 18eqeltrd 2890 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) ∈ No )
20 iffalse 4434 . . . . . 6 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
2120adantr 484 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
22 funmpt 6362 . . . . . . 7 Fun (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))
2322a1i 11 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → Fun (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
24 iotaex 6304 . . . . . . . . 9 (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ∈ V
25 eqid 2798 . . . . . . . . 9 (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))
2624, 25dmmpti 6464 . . . . . . . 8 dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}
27 ssel2 3910 . . . . . . . . . . . . . . . . 17 ((𝐴 No 𝑢𝐴) → 𝑢 No )
28 nodmon 33270 . . . . . . . . . . . . . . . . 17 (𝑢 No → dom 𝑢 ∈ On)
2927, 28syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 No 𝑢𝐴) → dom 𝑢 ∈ On)
30 onss 7485 . . . . . . . . . . . . . . . 16 (dom 𝑢 ∈ On → dom 𝑢 ⊆ On)
3129, 30syl 17 . . . . . . . . . . . . . . 15 ((𝐴 No 𝑢𝐴) → dom 𝑢 ⊆ On)
3231sseld 3914 . . . . . . . . . . . . . 14 ((𝐴 No 𝑢𝐴) → (𝑦 ∈ dom 𝑢𝑦 ∈ On))
3332adantrd 495 . . . . . . . . . . . . 13 ((𝐴 No 𝑢𝐴) → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ On))
3433rexlimdva 3243 . . . . . . . . . . . 12 (𝐴 No → (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ On))
3534abssdv 3996 . . . . . . . . . . 11 (𝐴 No → {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ On)
36 simplr 768 . . . . . . . . . . . . . . . . . . 19 (((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) → 𝑎𝑏)
3729adantlr 714 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) → dom 𝑢 ∈ On)
38 ontr1 6205 . . . . . . . . . . . . . . . . . . . 20 (dom 𝑢 ∈ On → ((𝑎𝑏𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
3937, 38syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) → ((𝑎𝑏𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
4036, 39mpand 694 . . . . . . . . . . . . . . . . . 18 (((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) → (𝑏 ∈ dom 𝑢𝑎 ∈ dom 𝑢))
4140adantrd 495 . . . . . . . . . . . . . . . . 17 (((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → 𝑎 ∈ dom 𝑢))
42 reseq1 5812 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎))
43 onelon 6184 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((dom 𝑢 ∈ On ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
4437, 43sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
45 suceloni 7508 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 ∈ On → suc 𝑏 ∈ On)
4644, 45syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑏 ∈ On)
47 simpllr 775 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → 𝑎𝑏)
48 eloni 6169 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑏 ∈ On → Ord 𝑏)
4944, 48syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → Ord 𝑏)
50 ordsucelsuc 7517 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Ord 𝑏 → (𝑎𝑏 ↔ suc 𝑎 ∈ suc 𝑏))
5149, 50syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → (𝑎𝑏 ↔ suc 𝑎 ∈ suc 𝑏))
5247, 51mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑎 ∈ suc 𝑏)
53 onelss 6201 . . . . . . . . . . . . . . . . . . . . . . . 24 (suc 𝑏 ∈ On → (suc 𝑎 ∈ suc 𝑏 → suc 𝑎 ⊆ suc 𝑏))
5446, 52, 53sylc 65 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑎 ⊆ suc 𝑏)
5554resabs1d 5849 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑢 ↾ suc 𝑎))
5654resabs1d 5849 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))
5755, 56eqeq12d 2814 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → (((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎) ↔ (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
5842, 57syl5ib 247 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
5958imim2d 57 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) → (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
6059ralimdv 3145 . . . . . . . . . . . . . . . . . 18 ((((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) ∧ 𝑏 ∈ dom 𝑢) → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) → ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
6160expimpd 457 . . . . . . . . . . . . . . . . 17 (((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
6241, 61jcad 516 . . . . . . . . . . . . . . . 16 (((𝐴 No 𝑎𝑏) ∧ 𝑢𝐴) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
6362reximdva 3233 . . . . . . . . . . . . . . 15 ((𝐴 No 𝑎𝑏) → (∃𝑢𝐴 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → ∃𝑢𝐴 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
6463expimpd 457 . . . . . . . . . . . . . 14 (𝐴 No → ((𝑎𝑏 ∧ ∃𝑢𝐴 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))) → ∃𝑢𝐴 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
65 vex 3444 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
66 eleq1w 2872 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → (𝑦 ∈ dom 𝑢𝑏 ∈ dom 𝑢))
67 suceq 6224 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
6867reseq2d 5818 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑏 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑏))
6967reseq2d 5818 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑏 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑏))
7068, 69eqeq12d 2814 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑏 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
7170imbi2d 344 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
7271ralbidv 3162 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
7366, 72anbi12d 633 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
7473rexbidv 3256 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏 → (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢𝐴 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
7565, 74elab 3615 . . . . . . . . . . . . . . 15 (𝑏 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢𝐴 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
7675anbi2i 625 . . . . . . . . . . . . . 14 ((𝑎𝑏𝑏 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) ↔ (𝑎𝑏 ∧ ∃𝑢𝐴 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
77 vex 3444 . . . . . . . . . . . . . . 15 𝑎 ∈ V
78 eleq1w 2872 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑎 → (𝑦 ∈ dom 𝑢𝑎 ∈ dom 𝑢))
79 suceq 6224 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑎 → suc 𝑦 = suc 𝑎)
8079reseq2d 5818 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑎 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑎))
8179reseq2d 5818 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑎 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑎))
8280, 81eqeq12d 2814 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑎 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
8382imbi2d 344 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑎 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
8483ralbidv 3162 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑎 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
8578, 84anbi12d 633 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑎 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
8685rexbidv 3256 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢𝐴 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
8777, 86elab 3615 . . . . . . . . . . . . . 14 (𝑎 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢𝐴 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
8864, 76, 873imtr4g 299 . . . . . . . . . . . . 13 (𝐴 No → ((𝑎𝑏𝑏 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
8988alrimivv 1929 . . . . . . . . . . . 12 (𝐴 No → ∀𝑎𝑏((𝑎𝑏𝑏 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
90 dftr2 5138 . . . . . . . . . . . 12 (Tr {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∀𝑎𝑏((𝑎𝑏𝑏 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
9189, 90sylibr 237 . . . . . . . . . . 11 (𝐴 No → Tr {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
92 dford5 33070 . . . . . . . . . . 11 (Ord {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ({𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ On ∧ Tr {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
9335, 91, 92sylanbrc 586 . . . . . . . . . 10 (𝐴 No → Ord {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
9493adantr 484 . . . . . . . . 9 ((𝐴 No 𝐴 ∈ V) → Ord {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
95 bdayfo 33295 . . . . . . . . . . . . . . 15 bday : No onto→On
96 fofun 6566 . . . . . . . . . . . . . . 15 ( bday : No onto→On → Fun bday )
9795, 96ax-mp 5 . . . . . . . . . . . . . 14 Fun bday
98 funimaexg 6410 . . . . . . . . . . . . . 14 ((Fun bday 𝐴 ∈ V) → ( bday 𝐴) ∈ V)
9997, 98mpan 689 . . . . . . . . . . . . 13 (𝐴 ∈ V → ( bday 𝐴) ∈ V)
10099uniexd 7448 . . . . . . . . . . . 12 (𝐴 ∈ V → ( bday 𝐴) ∈ V)
101100adantl 485 . . . . . . . . . . 11 ((𝐴 No 𝐴 ∈ V) → ( bday 𝐴) ∈ V)
102 simpl 486 . . . . . . . . . . . . . 14 ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ dom 𝑢)
103102reximi 3206 . . . . . . . . . . . . 13 (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → ∃𝑢𝐴 𝑦 ∈ dom 𝑢)
104103ss2abi 3994 . . . . . . . . . . . 12 {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ {𝑦 ∣ ∃𝑢𝐴 𝑦 ∈ dom 𝑢}
105 bdayval 33268 . . . . . . . . . . . . . . . . . . 19 (𝑢 No → ( bday 𝑢) = dom 𝑢)
10627, 105syl 17 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑢𝐴) → ( bday 𝑢) = dom 𝑢)
107 fofn 6567 . . . . . . . . . . . . . . . . . . . 20 ( bday : No onto→On → bday Fn No )
10895, 107ax-mp 5 . . . . . . . . . . . . . . . . . . 19 bday Fn No
109 fnfvima 6973 . . . . . . . . . . . . . . . . . . 19 (( bday Fn No 𝐴 No 𝑢𝐴) → ( bday 𝑢) ∈ ( bday 𝐴))
110108, 109mp3an1 1445 . . . . . . . . . . . . . . . . . 18 ((𝐴 No 𝑢𝐴) → ( bday 𝑢) ∈ ( bday 𝐴))
111106, 110eqeltrrd 2891 . . . . . . . . . . . . . . . . 17 ((𝐴 No 𝑢𝐴) → dom 𝑢 ∈ ( bday 𝐴))
112 elssuni 4830 . . . . . . . . . . . . . . . . 17 (dom 𝑢 ∈ ( bday 𝐴) → dom 𝑢 ( bday 𝐴))
113111, 112syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 No 𝑢𝐴) → dom 𝑢 ( bday 𝐴))
114113sseld 3914 . . . . . . . . . . . . . . 15 ((𝐴 No 𝑢𝐴) → (𝑦 ∈ dom 𝑢𝑦 ( bday 𝐴)))
115114rexlimdva 3243 . . . . . . . . . . . . . 14 (𝐴 No → (∃𝑢𝐴 𝑦 ∈ dom 𝑢𝑦 ( bday 𝐴)))
116115abssdv 3996 . . . . . . . . . . . . 13 (𝐴 No → {𝑦 ∣ ∃𝑢𝐴 𝑦 ∈ dom 𝑢} ⊆ ( bday 𝐴))
117116adantr 484 . . . . . . . . . . . 12 ((𝐴 No 𝐴 ∈ V) → {𝑦 ∣ ∃𝑢𝐴 𝑦 ∈ dom 𝑢} ⊆ ( bday 𝐴))
118104, 117sstrid 3926 . . . . . . . . . . 11 ((𝐴 No 𝐴 ∈ V) → {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ ( bday 𝐴))
119101, 118ssexd 5192 . . . . . . . . . 10 ((𝐴 No 𝐴 ∈ V) → {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ V)
120 elong 6167 . . . . . . . . . 10 ({𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ V → ({𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On ↔ Ord {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
121119, 120syl 17 . . . . . . . . 9 ((𝐴 No 𝐴 ∈ V) → ({𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On ↔ Ord {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
12294, 121mpbird 260 . . . . . . . 8 ((𝐴 No 𝐴 ∈ V) → {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On)
12326, 122eqeltrid 2894 . . . . . . 7 ((𝐴 No 𝐴 ∈ V) → dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ∈ On)
124123adantl 485 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ∈ On)
12525rnmpt 5791 . . . . . . . 8 ran (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = {𝑧 ∣ ∃𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))}
126 vex 3444 . . . . . . . . . . . 12 𝑔 ∈ V
127 eleq1w 2872 . . . . . . . . . . . . . 14 (𝑦 = 𝑔 → (𝑦 ∈ dom 𝑢𝑔 ∈ dom 𝑢))
128 suceq 6224 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑔 → suc 𝑦 = suc 𝑔)
129128reseq2d 5818 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑔 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑔))
130128reseq2d 5818 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑔 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑔))
131129, 130eqeq12d 2814 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑔 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))
132131imbi2d 344 . . . . . . . . . . . . . . 15 (𝑦 = 𝑔 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
133132ralbidv 3162 . . . . . . . . . . . . . 14 (𝑦 = 𝑔 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
134127, 133anbi12d 633 . . . . . . . . . . . . 13 (𝑦 = 𝑔 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))))
135134rexbidv 3256 . . . . . . . . . . . 12 (𝑦 = 𝑔 → (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))))
136126, 135elab 3615 . . . . . . . . . . 11 (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
137 eqid 2798 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑔) = (𝑢𝑔)
138 fvex 6658 . . . . . . . . . . . . . . . . . . . 20 (𝑢𝑔) ∈ V
139 eqeq2 2810 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = (𝑢𝑔) → ((𝑢𝑔) = 𝑥 ↔ (𝑢𝑔) = (𝑢𝑔)))
1401393anbi3d 1439 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (𝑢𝑔) → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = (𝑢𝑔))))
141138, 140spcev 3555 . . . . . . . . . . . . . . . . . . 19 ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = (𝑢𝑔)) → ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
142137, 141mp3an3 1447 . . . . . . . . . . . . . . . . . 18 ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
143142reximi 3206 . . . . . . . . . . . . . . . . 17 (∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑢𝐴𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
144 rexcom4 3212 . . . . . . . . . . . . . . . . 17 (∃𝑢𝐴𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ ∃𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
145143, 144sylib 221 . . . . . . . . . . . . . . . 16 (∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
146145adantl 485 . . . . . . . . . . . . . . 15 ((𝐴 No ∧ ∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
147 noprefixmo 33315 . . . . . . . . . . . . . . . 16 (𝐴 No → ∃*𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
148147adantr 484 . . . . . . . . . . . . . . 15 ((𝐴 No ∧ ∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃*𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
149 df-eu 2629 . . . . . . . . . . . . . . 15 (∃!𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ (∃𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ∧ ∃*𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))
150146, 148, 149sylanbrc 586 . . . . . . . . . . . . . 14 ((𝐴 No ∧ ∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃!𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))
151 vex 3444 . . . . . . . . . . . . . . 15 𝑧 ∈ V
152 eqeq2 2810 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → ((𝑢𝑔) = 𝑥 ↔ (𝑢𝑔) = 𝑧))
1531523anbi3d 1439 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧)))
154153rexbidv 3256 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ ∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧)))
155154iota2 6313 . . . . . . . . . . . . . . 15 ((𝑧 ∈ V ∧ ∃!𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) → (∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) ↔ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = 𝑧))
156151, 155mpan 689 . . . . . . . . . . . . . 14 (∃!𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) → (∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) ↔ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = 𝑧))
157150, 156syl 17 . . . . . . . . . . . . 13 ((𝐴 No ∧ ∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) ↔ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = 𝑧))
158 eqcom 2805 . . . . . . . . . . . . 13 ((℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = 𝑧𝑧 = (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))
159157, 158syl6bb 290 . . . . . . . . . . . 12 ((𝐴 No ∧ ∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) ↔ 𝑧 = (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
160 simprr3 1220 . . . . . . . . . . . . . . 15 ((𝐴 No ∧ (𝑢𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → (𝑢𝑔) = 𝑧)
16127adantrr 716 . . . . . . . . . . . . . . . . 17 ((𝐴 No ∧ (𝑢𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → 𝑢 No )
162 norn 33271 . . . . . . . . . . . . . . . . 17 (𝑢 No → ran 𝑢 ⊆ {1o, 2o})
163161, 162syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 No ∧ (𝑢𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → ran 𝑢 ⊆ {1o, 2o})
164 nofun 33269 . . . . . . . . . . . . . . . . . 18 (𝑢 No → Fun 𝑢)
165161, 164syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴 No ∧ (𝑢𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → Fun 𝑢)
166 simprr1 1218 . . . . . . . . . . . . . . . . 17 ((𝐴 No ∧ (𝑢𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → 𝑔 ∈ dom 𝑢)
167 fvelrn 6821 . . . . . . . . . . . . . . . . 17 ((Fun 𝑢𝑔 ∈ dom 𝑢) → (𝑢𝑔) ∈ ran 𝑢)
168165, 166, 167syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝐴 No ∧ (𝑢𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → (𝑢𝑔) ∈ ran 𝑢)
169163, 168sseldd 3916 . . . . . . . . . . . . . . 15 ((𝐴 No ∧ (𝑢𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → (𝑢𝑔) ∈ {1o, 2o})
170160, 169eqeltrrd 2891 . . . . . . . . . . . . . 14 ((𝐴 No ∧ (𝑢𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧))) → 𝑧 ∈ {1o, 2o})
171170rexlimdvaa 3244 . . . . . . . . . . . . 13 (𝐴 No → (∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) → 𝑧 ∈ {1o, 2o}))
172171adantr 484 . . . . . . . . . . . 12 ((𝐴 No ∧ ∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑧) → 𝑧 ∈ {1o, 2o}))
173159, 172sylbird 263 . . . . . . . . . . 11 ((𝐴 No ∧ ∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (𝑧 = (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) → 𝑧 ∈ {1o, 2o}))
174136, 173sylan2b 596 . . . . . . . . . 10 ((𝐴 No 𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → (𝑧 = (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) → 𝑧 ∈ {1o, 2o}))
175174rexlimdva 3243 . . . . . . . . 9 (𝐴 No → (∃𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) → 𝑧 ∈ {1o, 2o}))
176175abssdv 3996 . . . . . . . 8 (𝐴 No → {𝑧 ∣ ∃𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))} ⊆ {1o, 2o})
177125, 176eqsstrid 3963 . . . . . . 7 (𝐴 No → ran (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ⊆ {1o, 2o})
178177ad2antrl 727 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ran (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ⊆ {1o, 2o})
179 elno2 33274 . . . . . 6 ((𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ∈ No ↔ (Fun (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ∧ dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ∈ On ∧ ran (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ⊆ {1o, 2o}))
18023, 124, 178, 179syl3anbrc 1340 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) ∈ No )
18121, 180eqeltrd 2890 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) ∈ No )
18219, 181pm2.61ian 811 . . 3 ((𝐴 No 𝐴 ∈ V) → if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) ∈ No )
1832, 182eqeltrid 2894 . 2 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
1841, 183sylan2 595 1 ((𝐴 No 𝐴𝑉) → 𝑆 No )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wex 1781  wcel 2111  ∃*wmo 2596  ∃!weu 2628  {cab 2776  wral 3106  wrex 3107  ∃!wreu 3108  ∃*wrmo 3109  Vcvv 3441  cun 3879  wss 3881  ifcif 4425  {csn 4525  {cpr 4527  cop 4531   cuni 4800   class class class wbr 5030  cmpt 5110  Tr wtr 5136  dom cdm 5519  ran crn 5520  cres 5521  cima 5522  Ord word 6158  Oncon0 6159  suc csuc 6161  cio 6281  Fun wfun 6318   Fn wfn 6319  ontowfo 6322  cfv 6324  crio 7092  1oc1o 8078  2oc2o 8079   No csur 33260   <s cslt 33261   bday cbday 33262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-1o 8085  df-2o 8086  df-no 33263  df-slt 33264  df-bday 33265
This theorem is referenced by:  nosupbday  33318  nosupres  33320  nosupbnd1lem1  33321  nosupbnd1lem2  33322  nosupbnd1lem3  33323  nosupbnd1lem4  33324  nosupbnd1lem5  33325  nosupbnd1lem6  33326  nosupbnd2  33329  noetalem1  33330  noetalem2  33331  noetalem3  33332  noetalem4  33333
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