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Theorem noinfcbv 27777
Description: Change bound variables for surreal infimum. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypothesis
Ref Expression
noinfcbv.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
noinfcbv 𝑇 = if(∃𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎, ((𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎))))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑑   𝑔,𝑎,𝑢,𝑣   𝑥,𝑎,𝑦,𝐵,𝑏,𝑐,𝑑   𝑒,𝑏,𝐵   𝐵,𝑔   𝑢,𝑏,𝐵,𝑣   𝑥,𝑏,𝐵,𝑦   𝑐,𝑑,𝑒,𝑔,𝑢,𝑣   𝑥,𝑐,𝑦   𝑒,𝑑,𝑔,𝑢,𝑣   𝑥,𝑑,𝑦   𝑒,𝑔,𝑢,𝑣   𝑥,𝑒,𝑦   𝑢,𝑔,𝑣   𝑥,𝑔,𝑦   𝑣,𝑢   𝑥,𝑢,𝑦   𝑦,𝑣   𝑥,𝑦
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑣,𝑢,𝑒,𝑔,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem noinfcbv
StepHypRef Expression
1 noinfcbv.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 breq2 5152 . . . . . . 7 (𝑥 = 𝑎 → (𝑦 <s 𝑥𝑦 <s 𝑎))
32notbid 318 . . . . . 6 (𝑥 = 𝑎 → (¬ 𝑦 <s 𝑥 ↔ ¬ 𝑦 <s 𝑎))
43ralbidv 3176 . . . . 5 (𝑥 = 𝑎 → (∀𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ ∀𝑦𝐵 ¬ 𝑦 <s 𝑎))
5 breq1 5151 . . . . . . 7 (𝑦 = 𝑏 → (𝑦 <s 𝑎𝑏 <s 𝑎))
65notbid 318 . . . . . 6 (𝑦 = 𝑏 → (¬ 𝑦 <s 𝑎 ↔ ¬ 𝑏 <s 𝑎))
76cbvralvw 3235 . . . . 5 (∀𝑦𝐵 ¬ 𝑦 <s 𝑎 ↔ ∀𝑏𝐵 ¬ 𝑏 <s 𝑎)
84, 7bitrdi 287 . . . 4 (𝑥 = 𝑎 → (∀𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ ∀𝑏𝐵 ¬ 𝑏 <s 𝑎))
98cbvrexvw 3236 . . 3 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ ∃𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎)
108cbvriotavw 7398 . . . 4 (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = (𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎)
1110dmeqi 5918 . . . . . 6 dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = dom (𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎)
1211opeq1i 4881 . . . . 5 ⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩ = ⟨dom (𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎), 1o
1312sneqi 4642 . . . 4 {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩} = {⟨dom (𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}
1410, 13uneq12i 4176 . . 3 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = ((𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎), 1o⟩})
15 eleq1w 2822 . . . . . . . . 9 (𝑔 = 𝑐 → (𝑔 ∈ dom 𝑢𝑐 ∈ dom 𝑢))
16 suceq 6452 . . . . . . . . . . . . 13 (𝑔 = 𝑐 → suc 𝑔 = suc 𝑐)
1716reseq2d 6000 . . . . . . . . . . . 12 (𝑔 = 𝑐 → (𝑢 ↾ suc 𝑔) = (𝑢 ↾ suc 𝑐))
1816reseq2d 6000 . . . . . . . . . . . 12 (𝑔 = 𝑐 → (𝑣 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑐))
1917, 18eqeq12d 2751 . . . . . . . . . . 11 (𝑔 = 𝑐 → ((𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔) ↔ (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
2019imbi2d 340 . . . . . . . . . 10 (𝑔 = 𝑐 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
2120ralbidv 3176 . . . . . . . . 9 (𝑔 = 𝑐 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
22 fveqeq2 6916 . . . . . . . . 9 (𝑔 = 𝑐 → ((𝑢𝑔) = 𝑥 ↔ (𝑢𝑐) = 𝑥))
2315, 21, 223anbi123d 1435 . . . . . . . 8 (𝑔 = 𝑐 → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥)))
2423rexbidv 3177 . . . . . . 7 (𝑔 = 𝑐 → (∃𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥) ↔ ∃𝑢𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥)))
2524iotabidv 6547 . . . . . 6 (𝑔 = 𝑐 → (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = (℩𝑥𝑢𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥)))
26 eqeq2 2747 . . . . . . . . . 10 (𝑥 = 𝑎 → ((𝑢𝑐) = 𝑥 ↔ (𝑢𝑐) = 𝑎))
27263anbi3d 1441 . . . . . . . . 9 (𝑥 = 𝑎 → ((𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥) ↔ (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑎)))
2827rexbidv 3177 . . . . . . . 8 (𝑥 = 𝑎 → (∃𝑢𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥) ↔ ∃𝑢𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑎)))
29 dmeq 5917 . . . . . . . . . . 11 (𝑢 = 𝑑 → dom 𝑢 = dom 𝑑)
3029eleq2d 2825 . . . . . . . . . 10 (𝑢 = 𝑑 → (𝑐 ∈ dom 𝑢𝑐 ∈ dom 𝑑))
31 breq1 5151 . . . . . . . . . . . . . 14 (𝑢 = 𝑑 → (𝑢 <s 𝑣𝑑 <s 𝑣))
3231notbid 318 . . . . . . . . . . . . 13 (𝑢 = 𝑑 → (¬ 𝑢 <s 𝑣 ↔ ¬ 𝑑 <s 𝑣))
33 reseq1 5994 . . . . . . . . . . . . . 14 (𝑢 = 𝑑 → (𝑢 ↾ suc 𝑐) = (𝑑 ↾ suc 𝑐))
3433eqeq1d 2737 . . . . . . . . . . . . 13 (𝑢 = 𝑑 → ((𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)))
3532, 34imbi12d 344 . . . . . . . . . . . 12 (𝑢 = 𝑑 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
3635ralbidv 3176 . . . . . . . . . . 11 (𝑢 = 𝑑 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑣𝐵𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐))))
37 breq2 5152 . . . . . . . . . . . . . 14 (𝑣 = 𝑒 → (𝑑 <s 𝑣𝑑 <s 𝑒))
3837notbid 318 . . . . . . . . . . . . 13 (𝑣 = 𝑒 → (¬ 𝑑 <s 𝑣 ↔ ¬ 𝑑 <s 𝑒))
39 reseq1 5994 . . . . . . . . . . . . . 14 (𝑣 = 𝑒 → (𝑣 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))
4039eqeq2d 2746 . . . . . . . . . . . . 13 (𝑣 = 𝑒 → ((𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐) ↔ (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)))
4138, 40imbi12d 344 . . . . . . . . . . . 12 (𝑣 = 𝑒 → ((¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))))
4241cbvralvw 3235 . . . . . . . . . . 11 (∀𝑣𝐵𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)))
4336, 42bitrdi 287 . . . . . . . . . 10 (𝑢 = 𝑑 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ↔ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))))
44 fveq1 6906 . . . . . . . . . . 11 (𝑢 = 𝑑 → (𝑢𝑐) = (𝑑𝑐))
4544eqeq1d 2737 . . . . . . . . . 10 (𝑢 = 𝑑 → ((𝑢𝑐) = 𝑎 ↔ (𝑑𝑐) = 𝑎))
4630, 43, 453anbi123d 1435 . . . . . . . . 9 (𝑢 = 𝑑 → ((𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑎) ↔ (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎)))
4746cbvrexvw 3236 . . . . . . . 8 (∃𝑢𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑎) ↔ ∃𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎))
4828, 47bitrdi 287 . . . . . . 7 (𝑥 = 𝑎 → (∃𝑢𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥) ↔ ∃𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎)))
4948cbviotavw 6524 . . . . . 6 (℩𝑥𝑢𝐵 (𝑐 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑐) = (𝑣 ↾ suc 𝑐)) ∧ (𝑢𝑐) = 𝑥)) = (℩𝑎𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎))
5025, 49eqtrdi 2791 . . . . 5 (𝑔 = 𝑐 → (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) = (℩𝑎𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎)))
5150cbvmptv 5261 . . . 4 (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))) = (𝑐 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑎𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎)))
52 eleq1w 2822 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑦 ∈ dom 𝑢𝑏 ∈ dom 𝑢))
53 suceq 6452 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
5453reseq2d 6000 . . . . . . . . . . . 12 (𝑦 = 𝑏 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑏))
5553reseq2d 6000 . . . . . . . . . . . 12 (𝑦 = 𝑏 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑏))
5654, 55eqeq12d 2751 . . . . . . . . . . 11 (𝑦 = 𝑏 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
5756imbi2d 340 . . . . . . . . . 10 (𝑦 = 𝑏 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
5857ralbidv 3176 . . . . . . . . 9 (𝑦 = 𝑏 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
5952, 58anbi12d 632 . . . . . . . 8 (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)))))
6059rexbidv 3177 . . . . . . 7 (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)))))
6129eleq2d 2825 . . . . . . . . 9 (𝑢 = 𝑑 → (𝑏 ∈ dom 𝑢𝑏 ∈ dom 𝑑))
62 reseq1 5994 . . . . . . . . . . . . 13 (𝑢 = 𝑑 → (𝑢 ↾ suc 𝑏) = (𝑑 ↾ suc 𝑏))
6362eqeq1d 2737 . . . . . . . . . . . 12 (𝑢 = 𝑑 → ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) ↔ (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
6432, 63imbi12d 344 . . . . . . . . . . 11 (𝑢 = 𝑑 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ (¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
6564ralbidv 3176 . . . . . . . . . 10 (𝑢 = 𝑑 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ ∀𝑣𝐵𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
66 reseq1 5994 . . . . . . . . . . . . 13 (𝑣 = 𝑒 → (𝑣 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏))
6766eqeq2d 2746 . . . . . . . . . . . 12 (𝑣 = 𝑒 → ((𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) ↔ (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))
6838, 67imbi12d 344 . . . . . . . . . . 11 (𝑣 = 𝑒 → ((¬ 𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏))))
6968cbvralvw 3235 . . . . . . . . . 10 (∀𝑣𝐵𝑑 <s 𝑣 → (𝑑 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))
7065, 69bitrdi 287 . . . . . . . . 9 (𝑢 = 𝑑 → (∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) ↔ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏))))
7161, 70anbi12d 632 . . . . . . . 8 (𝑢 = 𝑑 → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) ↔ (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))))
7271cbvrexvw 3236 . . . . . . 7 (∃𝑢𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) ↔ ∃𝑑𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏))))
7360, 72bitrdi 287 . . . . . 6 (𝑦 = 𝑏 → (∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑑𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))))
7473cbvabv 2810 . . . . 5 {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} = {𝑏 ∣ ∃𝑑𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))}
7574mpteq1i 5244 . . . 4 (𝑐 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑎𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎))) = (𝑐 ∈ {𝑏 ∣ ∃𝑑𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎)))
7651, 75eqtri 2763 . . 3 (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))) = (𝑐 ∈ {𝑏 ∣ ∃𝑑𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎)))
779, 14, 76ifbieq12i 4558 . 2 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)))) = if(∃𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎, ((𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎))))
781, 77eqtri 2763 1 𝑇 = if(∃𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎, ((𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  cun 3961  ifcif 4531  {csn 4631  cop 4637   class class class wbr 5148  cmpt 5231  dom cdm 5689  cres 5691  suc csuc 6388  cio 6514  cfv 6563  crio 7387  1oc1o 8498   <s cslt 27700
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-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-dm 5699  df-res 5701  df-suc 6392  df-iota 6516  df-fv 6571  df-riota 7388
This theorem is referenced by:  noeta  27803
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