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Theorem nosupres 27617
Description: A restriction law for surreal supremum when there is no maximum. (Contributed by Scott Fenton, 5-Dec-2021.)
Hypothesis
Ref Expression
nosupres.1 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
Assertion
Ref Expression
nosupres ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))
Distinct variable groups:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦   𝑢,𝐺,𝑣,𝑦   𝑥,𝑈,𝑢,𝑣
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑈(𝑦,𝑔)   𝐺(𝑥,𝑔)
Proof of Theorem nosupres
Dummy variables 𝑎 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmres 5963 . . . 4 dom (𝑆 ↾ suc 𝐺) = (suc 𝐺 ∩ dom 𝑆)
2 nosupres.1 . . . . . . . . 9 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
32nosupno 27613 . . . . . . . 8 ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
433ad2ant2 1134 . . . . . . 7 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝑆 ∈ No )
5 nodmord 27563 . . . . . . 7 (𝑆 ∈ No → Ord dom 𝑆)
64, 5syl 17 . . . . . 6 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → Ord dom 𝑆)
7 dmeq 5846 . . . . . . . . . . . . . 14 (𝑝 = 𝑈 → dom 𝑝 = dom 𝑈)
87eleq2d 2814 . . . . . . . . . . . . 13 (𝑝 = 𝑈 → (𝐺 ∈ dom 𝑝 ↔ 𝐺 ∈ dom 𝑈))
9 breq2 5096 . . . . . . . . . . . . . . . 16 (𝑝 = 𝑈 → (𝑣 <s 𝑝 ↔ 𝑣 <s 𝑈))
109notbid 318 . . . . . . . . . . . . . . 15 (𝑝 = 𝑈 → (¬ 𝑣 <s 𝑝 ↔ ¬ 𝑣 <s 𝑈))
11 reseq1 5924 . . . . . . . . . . . . . . . 16 (𝑝 = 𝑈 → (𝑝 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))
1211eqeq1d 2731 . . . . . . . . . . . . . . 15 (𝑝 = 𝑈 → ((𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
1310, 12imbi12d 344 . . . . . . . . . . . . . 14 (𝑝 = 𝑈 → ((¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
1413ralbidv 3152 . . . . . . . . . . . . 13 (𝑝 = 𝑈 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
158, 14anbi12d 632 . . . . . . . . . . . 12 (𝑝 = 𝑈 → ((𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))) ↔ (𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))))
1615rspcev 3577 . . . . . . . . . . 11 ((𝑈 ∈ 𝐴 ∧ (𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
17163impb 1114 . . . . . . . . . 10 ((𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))) → ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
18 dmeq 5846 . . . . . . . . . . . . 13 (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
1918eleq2d 2814 . . . . . . . . . . . 12 (𝑢 = 𝑝 → (𝐺 ∈ dom 𝑢 ↔ 𝐺 ∈ dom 𝑝))
20 breq2 5096 . . . . . . . . . . . . . . 15 (𝑢 = 𝑝 → (𝑣 <s 𝑢 ↔ 𝑣 <s 𝑝))
2120notbid 318 . . . . . . . . . . . . . 14 (𝑢 = 𝑝 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝))
22 reseq1 5924 . . . . . . . . . . . . . . 15 (𝑢 = 𝑝 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
2322eqeq1d 2731 . . . . . . . . . . . . . 14 (𝑢 = 𝑝 → ((𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
2421, 23imbi12d 344 . . . . . . . . . . . . 13 (𝑢 = 𝑝 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
2524ralbidv 3152 . . . . . . . . . . . 12 (𝑢 = 𝑝 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
2619, 25anbi12d 632 . . . . . . . . . . 11 (𝑢 = 𝑝 → ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))) ↔ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))))
2726cbvrexvw 3208 . . . . . . . . . 10 (∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))) ↔ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
2817, 27sylibr 234 . . . . . . . . 9 ((𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))) → ∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
29 eleq1 2816 . . . . . . . . . . . . 13 (𝑦 = 𝐺 → (𝑦 ∈ dom 𝑢 ↔ 𝐺 ∈ dom 𝑢))
30 suceq 6375 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐺 → suc 𝑦 = suc 𝐺)
3130reseq2d 5930 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐺 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝐺))
3230reseq2d 5930 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐺 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝐺))
3331, 32eqeq12d 2745 . . . . . . . . . . . . . . 15 (𝑦 = 𝐺 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
3433imbi2d 340 . . . . . . . . . . . . . 14 (𝑦 = 𝐺 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
3534ralbidv 3152 . . . . . . . . . . . . 13 (𝑦 = 𝐺 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
3629, 35anbi12d 632 . . . . . . . . . . . 12 (𝑦 = 𝐺 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))))
3736rexbidv 3153 . . . . . . . . . . 11 (𝑦 = 𝐺 → (∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))))
3837elabg 3632 . . . . . . . . . 10 (𝐺 ∈ dom 𝑈 → (𝐺 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))))
39383ad2ant2 1134 . . . . . . . . 9 ((𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))) → (𝐺 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))))
4028, 39mpbird 257 . . . . . . . 8 ((𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))) → 𝐺 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
41403ad2ant3 1135 . . . . . . 7 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝐺 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
42 iffalse 4485 . . . . . . . . . . 11 (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
432, 42eqtrid 2776 . . . . . . . . . 10 (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → 𝑆 = (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
4443dmeqd 5848 . . . . . . . . 9 (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
45 iotaex 6458 . . . . . . . . . 10 (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) ∈ V
46 eqid 2729 . . . . . . . . . 10 (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))
4745, 46dmmpti 6626 . . . . . . . . 9 dom (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) = {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}
4844, 47eqtrdi 2780 . . . . . . . 8 (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
49483ad2ant1 1133 . . . . . . 7 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → dom 𝑆 = {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
5041, 49eleqtrrd 2831 . . . . . 6 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝐺 ∈ dom 𝑆)
51 ordsucss 7751 . . . . . 6 (Ord dom 𝑆 → (𝐺 ∈ dom 𝑆 → suc 𝐺 ⊆ dom 𝑆))
526, 50, 51sylc 65 . . . . 5 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → suc 𝐺 ⊆ dom 𝑆)
53 dfss2 3921 . . . . 5 (suc 𝐺 ⊆ dom 𝑆 ↔ (suc 𝐺 ∩ dom 𝑆) = suc 𝐺)
5452, 53sylib 218 . . . 4 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (suc 𝐺 ∩ dom 𝑆) = suc 𝐺)
551, 54eqtrid 2776 . . 3 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → dom (𝑆 ↾ suc 𝐺) = suc 𝐺)
56 dmres 5963 . . . 4 dom (𝑈 ↾ suc 𝐺) = (suc 𝐺 ∩ dom 𝑈)
57 simp2l 1200 . . . . . . . 8 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝐴 ⊆ No )
58 simp31 1210 . . . . . . . 8 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝑈 ∈ 𝐴)
5957, 58sseldd 3936 . . . . . . 7 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝑈 ∈ No )
60 nodmord 27563 . . . . . . 7 (𝑈 ∈ No → Ord dom 𝑈)
6159, 60syl 17 . . . . . 6 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → Ord dom 𝑈)
62 simp32 1211 . . . . . 6 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝐺 ∈ dom 𝑈)
63 ordsucss 7751 . . . . . 6 (Ord dom 𝑈 → (𝐺 ∈ dom 𝑈 → suc 𝐺 ⊆ dom 𝑈))
6461, 62, 63sylc 65 . . . . 5 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → suc 𝐺 ⊆ dom 𝑈)
65 dfss2 3921 . . . . 5 (suc 𝐺 ⊆ dom 𝑈 ↔ (suc 𝐺 ∩ dom 𝑈) = suc 𝐺)
6664, 65sylib 218 . . . 4 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (suc 𝐺 ∩ dom 𝑈) = suc 𝐺)
6756, 66eqtrid 2776 . . 3 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → dom (𝑈 ↾ suc 𝐺) = suc 𝐺)
6855, 67eqtr4d 2767 . 2 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → dom (𝑆 ↾ suc 𝐺) = dom (𝑈 ↾ suc 𝐺))
6955eleq2d 2814 . . . 4 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑎 ∈ dom (𝑆 ↾ suc 𝐺) ↔ 𝑎 ∈ suc 𝐺))
70 simpl1 1192 . . . . . . 7 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
71 simpl2 1193 . . . . . . 7 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
72 simpl31 1255 . . . . . . 7 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → 𝑈 ∈ 𝐴)
7364sselda 3935 . . . . . . 7 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → 𝑎 ∈ dom 𝑈)
74 nodmon 27560 . . . . . . . . . . . . . 14 (𝑈 ∈ No → dom 𝑈 ∈ On)
7559, 74syl 17 . . . . . . . . . . . . 13 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → dom 𝑈 ∈ On)
76 onelon 6332 . . . . . . . . . . . . 13 ((dom 𝑈 ∈ On ∧ 𝐺 ∈ dom 𝑈) → 𝐺 ∈ On)
7775, 62, 76syl2anc 584 . . . . . . . . . . . 12 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → 𝐺 ∈ On)
78 eloni 6317 . . . . . . . . . . . 12 (𝐺 ∈ On → Ord 𝐺)
7977, 78syl 17 . . . . . . . . . . 11 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → Ord 𝐺)
80 ordsuc 7747 . . . . . . . . . . 11 (Ord 𝐺 ↔ Ord suc 𝐺)
8179, 80sylib 218 . . . . . . . . . 10 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → Ord suc 𝐺)
82 ordsucss 7751 . . . . . . . . . 10 (Ord suc 𝐺 → (𝑎 ∈ suc 𝐺 → suc 𝑎 ⊆ suc 𝐺))
8381, 82syl 17 . . . . . . . . 9 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑎 ∈ suc 𝐺 → suc 𝑎 ⊆ suc 𝐺))
8483imp 406 . . . . . . . 8 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → suc 𝑎 ⊆ suc 𝐺)
85 simpl33 1257 . . . . . . . 8 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
86 reseq1 5924 . . . . . . . . . . 11 ((𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) → ((𝑈 ↾ suc 𝐺) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝐺) ↾ suc 𝑎))
87 resabs1 5957 . . . . . . . . . . . 12 (suc 𝑎 ⊆ suc 𝐺 → ((𝑈 ↾ suc 𝐺) ↾ suc 𝑎) = (𝑈 ↾ suc 𝑎))
88 resabs1 5957 . . . . . . . . . . . 12 (suc 𝑎 ⊆ suc 𝐺 → ((𝑣 ↾ suc 𝐺) ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))
8987, 88eqeq12d 2745 . . . . . . . . . . 11 (suc 𝑎 ⊆ suc 𝐺 → (((𝑈 ↾ suc 𝐺) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝐺) ↾ suc 𝑎) ↔ (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
9086, 89imbitrid 244 . . . . . . . . . 10 (suc 𝑎 ⊆ suc 𝐺 → ((𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) → (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
9190imim2d 57 . . . . . . . . 9 (suc 𝑎 ⊆ suc 𝐺 → ((¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) → (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
9291ralimdv 3143 . . . . . . . 8 (suc 𝑎 ⊆ suc 𝐺 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) → ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
9384, 85, 92sylc 65 . . . . . . 7 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
942nosupfv 27616 . . . . . . 7 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝑎 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))) → (𝑆‘𝑎) = (𝑈‘𝑎))
9570, 71, 72, 73, 93, 94syl113anc 1384 . . . . . 6 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → (𝑆‘𝑎) = (𝑈‘𝑎))
96 simpr 484 . . . . . . 7 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → 𝑎 ∈ suc 𝐺)
9796fvresd 6842 . . . . . 6 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ((𝑆 ↾ suc 𝐺)‘𝑎) = (𝑆‘𝑎))
9896fvresd 6842 . . . . . 6 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ((𝑈 ↾ suc 𝐺)‘𝑎) = (𝑈‘𝑎))
9995, 97, 983eqtr4d 2774 . . . . 5 (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) ∧ 𝑎 ∈ suc 𝐺) → ((𝑆 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎))
10099ex 412 . . . 4 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑎 ∈ suc 𝐺 → ((𝑆 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎)))
10169, 100sylbid 240 . . 3 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑎 ∈ dom (𝑆 ↾ suc 𝐺) → ((𝑆 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎)))
102101ralrimiv 3120 . 2 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → ∀𝑎 ∈ dom (𝑆 ↾ suc 𝐺)((𝑆 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎))
103 nofun 27559 . . . 4 (𝑆 ∈ No → Fun 𝑆)
104 funres 6524 . . . 4 (Fun 𝑆 → Fun (𝑆 ↾ suc 𝐺))
1054, 103, 1043syl 18 . . 3 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → Fun (𝑆 ↾ suc 𝐺))
106 nofun 27559 . . . 4 (𝑈 ∈ No → Fun 𝑈)
107 funres 6524 . . . 4 (Fun 𝑈 → Fun (𝑈 ↾ suc 𝐺))
10859, 106, 1073syl 18 . . 3 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → Fun (𝑈 ↾ suc 𝐺))
109 eqfunfv 6970 . . 3 ((Fun (𝑆 ↾ suc 𝐺) ∧ Fun (𝑈 ↾ suc 𝐺)) → ((𝑆 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺) ↔ (dom (𝑆 ↾ suc 𝐺) = dom (𝑈 ↾ suc 𝐺) ∧ ∀𝑎 ∈ dom (𝑆 ↾ suc 𝐺)((𝑆 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎))))
110105, 108, 109syl2anc 584 . 2 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → ((𝑆 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺) ↔ (dom (𝑆 ↾ suc 𝐺) = dom (𝑈 ↾ suc 𝐺) ∧ ∀𝑎 ∈ dom (𝑆 ↾ suc 𝐺)((𝑆 ↾ suc 𝐺)‘𝑎) = ((𝑈 ↾ suc 𝐺)‘𝑎))))
11168, 102, 110mpbir2and 713 1 ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ifcif 4476  {csn 4577  ⟨cop 4583   class class class wbr 5092   ↦ cmpt 5173  dom cdm 5619   ↾ cres 5621  Ord word 6306  Oncon0 6307  suc csuc 6309  ℩cio 6436  Fun wfun 6476  ‘cfv 6482  ℩crio 7305  2oc2o 8382   No csur 27549   <s cslt 27550
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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fo 6488  df-fv 6490  df-riota 7306  df-1o 8388  df-2o 8389  df-no 27552  df-slt 27553  df-bday 27554
This theorem is referenced by:  nosupbnd1lem1  27618  nosupbnd2  27626
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