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Theorem nosupcbv 33470
 Description: Lemma to change bound variables in a surreal supremum. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypothesis
Ref Expression
nosupcbv.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
nosupcbv 𝑆 = if(∃𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏, ((𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2o⟩}), (𝑐 ∈ {𝑑 ∣ ∃𝑒𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎𝑒𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎))))
Distinct variable groups:   𝐴,𝑎,𝑏   𝑎,𝑐,𝐴   𝐴,𝑑   𝑒,𝑎,𝐴   𝐴,𝑓   𝑔,𝑎,𝐴,𝑢,𝑣   𝑥,𝑎,𝐴,𝑦,𝑏   𝑐,𝑑,𝑒,𝑓   𝑔,𝑐,𝑢,𝑣   𝑥,𝑐,𝑦   𝑒,𝑑,𝑓,𝑢,𝑣,𝑦   𝑒,𝑔,𝑢,𝑣   𝑥,𝑒,𝑦   𝑓,𝑔,𝑢,𝑣   𝑥,𝑓,𝑦   𝑢,𝑔,𝑣   𝑥,𝑔,𝑦   𝑣,𝑢   𝑥,𝑢,𝑦   𝑦,𝑣   𝑥,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑣,𝑢,𝑒,𝑓,𝑔,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem nosupcbv
StepHypRef Expression
1 nosupcbv.1 . 2 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))))
2 breq1 5035 . . . . . . 7 (𝑥 = 𝑎 → (𝑥 <s 𝑦𝑎 <s 𝑦))
32notbid 321 . . . . . 6 (𝑥 = 𝑎 → (¬ 𝑥 <s 𝑦 ↔ ¬ 𝑎 <s 𝑦))
43ralbidv 3126 . . . . 5 (𝑥 = 𝑎 → (∀𝑦𝐴 ¬ 𝑥 <s 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑎 <s 𝑦))
5 breq2 5036 . . . . . . 7 (𝑦 = 𝑏 → (𝑎 <s 𝑦𝑎 <s 𝑏))
65notbid 321 . . . . . 6 (𝑦 = 𝑏 → (¬ 𝑎 <s 𝑦 ↔ ¬ 𝑎 <s 𝑏))
76cbvralvw 3361 . . . . 5 (∀𝑦𝐴 ¬ 𝑎 <s 𝑦 ↔ ∀𝑏𝐴 ¬ 𝑎 <s 𝑏)
84, 7bitrdi 290 . . . 4 (𝑥 = 𝑎 → (∀𝑦𝐴 ¬ 𝑥 <s 𝑦 ↔ ∀𝑏𝐴 ¬ 𝑎 <s 𝑏))
98cbvrexvw 3362 . . 3 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ↔ ∃𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏)
108cbvriotavw 7118 . . . 4 (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏)
1110dmeqi 5744 . . . . . 6 dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏)
1211opeq1i 4766 . . . . 5 ⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩ = ⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2o
1312sneqi 4533 . . . 4 {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩} = {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2o⟩}
1410, 13uneq12i 4066 . . 3 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) = ((𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2o⟩})
15 eleq1w 2834 . . . . . . . . 9 (𝑔 = 𝑐 → (𝑔 ∈ dom 𝑢𝑐 ∈ dom 𝑢))
16 suceq 6234 . . . . . . . . . . . . 13 (𝑔 = 𝑐 → suc 𝑔 = suc 𝑐)
1716reseq2d 5823 . . . . . . . . . . . 12 (𝑔 = 𝑐 → (𝑢 ↾ suc 𝑔) = (𝑢 ↾ suc 𝑐))
1816reseq2d 5823 . . . . . . . . . . . 12 (𝑔 = 𝑐 → (𝑣 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑐))
1917, 18eqeq12d 2774 . . . . . . . . . . 11 (𝑔 = 𝑐 → ((𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔) ↔ (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
2019imbi2d 344 . . . . . . . . . 10 (𝑔 = 𝑐 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
2120ralbidv 3126 . . . . . . . . 9 (𝑔 = 𝑐 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
22 fveqeq2 6667 . . . . . . . . 9 (𝑔 = 𝑐 → ((𝑢𝑔) = 𝑥 ↔ (𝑢𝑐) = 𝑥))
2315, 21, 223anbi123d 1433 . . . . . . . 8 (𝑔 = 𝑐 → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥)))
2423rexbidv 3221 . . . . . . 7 (𝑔 = 𝑐 → (∃𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ ∃𝑢𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥)))
2524iotabidv 6319 . . . . . 6 (𝑔 = 𝑐 → (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = (℩𝑥𝑢𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥)))
26 eqeq2 2770 . . . . . . . . . 10 (𝑥 = 𝑎 → ((𝑢𝑐) = 𝑥 ↔ (𝑢𝑐) = 𝑎))
27263anbi3d 1439 . . . . . . . . 9 (𝑥 = 𝑎 → ((𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥) ↔ (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑎)))
2827rexbidv 3221 . . . . . . . 8 (𝑥 = 𝑎 → (∃𝑢𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥) ↔ ∃𝑢𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑎)))
29 dmeq 5743 . . . . . . . . . . 11 (𝑢 = 𝑒 → dom 𝑢 = dom 𝑒)
3029eleq2d 2837 . . . . . . . . . 10 (𝑢 = 𝑒 → (𝑐 ∈ dom 𝑢𝑐 ∈ dom 𝑒))
31 breq2 5036 . . . . . . . . . . . . . 14 (𝑢 = 𝑒 → (𝑣 <s 𝑢𝑣 <s 𝑒))
3231notbid 321 . . . . . . . . . . . . 13 (𝑢 = 𝑒 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑒))
33 reseq1 5817 . . . . . . . . . . . . . 14 (𝑢 = 𝑒 → (𝑢 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))
3433eqeq1d 2760 . . . . . . . . . . . . 13 (𝑢 = 𝑒 → ((𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
3532, 34imbi12d 348 . . . . . . . . . . . 12 (𝑢 = 𝑒 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
3635ralbidv 3126 . . . . . . . . . . 11 (𝑢 = 𝑒 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑣𝐴𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
37 breq1 5035 . . . . . . . . . . . . . 14 (𝑣 = 𝑓 → (𝑣 <s 𝑒𝑓 <s 𝑒))
3837notbid 321 . . . . . . . . . . . . 13 (𝑣 = 𝑓 → (¬ 𝑣 <s 𝑒 ↔ ¬ 𝑓 <s 𝑒))
39 reseq1 5817 . . . . . . . . . . . . . 14 (𝑣 = 𝑓 → (𝑣 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐))
4039eqeq2d 2769 . . . . . . . . . . . . 13 (𝑣 = 𝑓 → ((𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)))
4138, 40imbi12d 348 . . . . . . . . . . . 12 (𝑣 = 𝑓 → ((¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐))))
4241cbvralvw 3361 . . . . . . . . . . 11 (∀𝑣𝐴𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)))
4336, 42bitrdi 290 . . . . . . . . . 10 (𝑢 = 𝑒 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐))))
44 fveq1 6657 . . . . . . . . . . 11 (𝑢 = 𝑒 → (𝑢𝑐) = (𝑒𝑐))
4544eqeq1d 2760 . . . . . . . . . 10 (𝑢 = 𝑒 → ((𝑢𝑐) = 𝑎 ↔ (𝑒𝑐) = 𝑎))
4630, 43, 453anbi123d 1433 . . . . . . . . 9 (𝑢 = 𝑒 → ((𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑎) ↔ (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎)))
4746cbvrexvw 3362 . . . . . . . 8 (∃𝑢𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑎) ↔ ∃𝑒𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎))
4828, 47bitrdi 290 . . . . . . 7 (𝑥 = 𝑎 → (∃𝑢𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥) ↔ ∃𝑒𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎)))
4948cbviotavw 6302 . . . . . 6 (℩𝑥𝑢𝐴 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥)) = (℩𝑎𝑒𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎))
5025, 49eqtrdi 2809 . . . . 5 (𝑔 = 𝑐 → (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = (℩𝑎𝑒𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎)))
5150cbvmptv 5135 . . . 4 (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))) = (𝑐 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑎𝑒𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎)))
52 eleq1w 2834 . . . . . . . . 9 (𝑦 = 𝑑 → (𝑦 ∈ dom 𝑢𝑑 ∈ dom 𝑢))
53 suceq 6234 . . . . . . . . . . . . 13 (𝑦 = 𝑑 → suc 𝑦 = suc 𝑑)
5453reseq2d 5823 . . . . . . . . . . . 12 (𝑦 = 𝑑 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑑))
5553reseq2d 5823 . . . . . . . . . . . 12 (𝑦 = 𝑑 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑑))
5654, 55eqeq12d 2774 . . . . . . . . . . 11 (𝑦 = 𝑑 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)))
5756imbi2d 344 . . . . . . . . . 10 (𝑦 = 𝑑 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))))
5857ralbidv 3126 . . . . . . . . 9 (𝑦 = 𝑑 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))))
5952, 58anbi12d 633 . . . . . . . 8 (𝑦 = 𝑑 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑑 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)))))
6059rexbidv 3221 . . . . . . 7 (𝑦 = 𝑑 → (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢𝐴 (𝑑 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)))))
6129eleq2d 2837 . . . . . . . . 9 (𝑢 = 𝑒 → (𝑑 ∈ dom 𝑢𝑑 ∈ dom 𝑒))
62 reseq1 5817 . . . . . . . . . . . . 13 (𝑢 = 𝑒 → (𝑢 ↾ suc 𝑑) = (𝑒 ↾ suc 𝑑))
6362eqeq1d 2760 . . . . . . . . . . . 12 (𝑢 = 𝑒 → ((𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑) ↔ (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)))
6432, 63imbi12d 348 . . . . . . . . . . 11 (𝑢 = 𝑒 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ (¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))))
6564ralbidv 3126 . . . . . . . . . 10 (𝑢 = 𝑒 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ ∀𝑣𝐴𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))))
66 reseq1 5817 . . . . . . . . . . . . 13 (𝑣 = 𝑓 → (𝑣 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑))
6766eqeq2d 2769 . . . . . . . . . . . 12 (𝑣 = 𝑓 → ((𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑) ↔ (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))
6838, 67imbi12d 348 . . . . . . . . . . 11 (𝑣 = 𝑓 → ((¬ 𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑))))
6968cbvralvw 3361 . . . . . . . . . 10 (∀𝑣𝐴𝑣 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))
7065, 69bitrdi 290 . . . . . . . . 9 (𝑢 = 𝑒 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑)) ↔ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑))))
7161, 70anbi12d 633 . . . . . . . 8 (𝑢 = 𝑒 → ((𝑑 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))) ↔ (𝑑 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))))
7271cbvrexvw 3362 . . . . . . 7 (∃𝑢𝐴 (𝑑 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑑) = (𝑣 ↾ suc 𝑑))) ↔ ∃𝑒𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑))))
7360, 72bitrdi 290 . . . . . 6 (𝑦 = 𝑑 → (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑒𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))))
7473cbvabv 2826 . . . . 5 {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} = {𝑑 ∣ ∃𝑒𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))}
7574mpteq1i 5122 . . . 4 (𝑐 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑎𝑒𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎))) = (𝑐 ∈ {𝑑 ∣ ∃𝑒𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎𝑒𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎)))
7651, 75eqtri 2781 . . 3 (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))) = (𝑐 ∈ {𝑑 ∣ ∃𝑒𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎𝑒𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎)))
779, 14, 76ifbieq12i 4447 . 2 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)))) = if(∃𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏, ((𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2o⟩}), (𝑐 ∈ {𝑑 ∣ ∃𝑒𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎𝑒𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎))))
781, 77eqtri 2781 1 𝑆 = if(∃𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏, ((𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2o⟩}), (𝑐 ∈ {𝑑 ∣ ∃𝑒𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎𝑒𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓𝐴𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒𝑐) = 𝑎))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071   ∪ cun 3856  ifcif 4420  {csn 4522  ⟨cop 4528   class class class wbr 5032   ↦ cmpt 5112  dom cdm 5524   ↾ cres 5526  suc csuc 6171  ℩cio 6292  ‘cfv 6335  ℩crio 7107  2oc2o 8106
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