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Theorem eqscut2 27167
Description: Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
Assertion
Ref Expression
eqscut2 ((𝐿 <<s 𝑅𝑋 No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦)))))
Distinct variable groups:   𝑦,𝐿   𝑦,𝑅   𝑦,𝑋

Proof of Theorem eqscut2
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqscut 27166 . 2 ((𝐿 <<s 𝑅𝑋 No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday 𝑋) = ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))))
2 eqss 3964 . . . . 5 (( bday 𝑋) = ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ (( bday 𝑋) ⊆ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ∧ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ ( bday 𝑋)))
3 sneq 4601 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → {𝑥} = {𝑋})
43breq2d 5122 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
53breq1d 5120 . . . . . . . . . . . 12 (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
64, 5anbi12d 632 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
76elrab3 3651 . . . . . . . . . 10 (𝑋 No → (𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
87adantl 483 . . . . . . . . 9 ((𝐿 <<s 𝑅𝑋 No ) → (𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
98biimpar 479 . . . . . . . 8 (((𝐿 <<s 𝑅𝑋 No ) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → 𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})
10 bdayfn 27135 . . . . . . . . 9 bday Fn No
11 ssrab2 4042 . . . . . . . . 9 {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
12 fnfvima 7188 . . . . . . . . 9 (( bday Fn No ∧ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No 𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → ( bday 𝑋) ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
1310, 11, 12mp3an12 1452 . . . . . . . 8 (𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} → ( bday 𝑋) ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
14 intss1 4929 . . . . . . . 8 (( bday 𝑋) ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ ( bday 𝑋))
159, 13, 143syl 18 . . . . . . 7 (((𝐿 <<s 𝑅𝑋 No ) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ ( bday 𝑋))
1615biantrud 533 . . . . . 6 (((𝐿 <<s 𝑅𝑋 No ) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → (( bday 𝑋) ⊆ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ (( bday 𝑋) ⊆ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ∧ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ ( bday 𝑋))))
17 ssint 4930 . . . . . . 7 (( bday 𝑋) ⊆ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∀𝑧 ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})( bday 𝑋) ⊆ 𝑧)
18 fvelimab 6919 . . . . . . . . . . . . . 14 (( bday Fn No ∧ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No ) → (𝑧 ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∃𝑦 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ( bday 𝑦) = 𝑧))
1910, 11, 18mp2an 691 . . . . . . . . . . . . 13 (𝑧 ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∃𝑦 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ( bday 𝑦) = 𝑧)
20 sneq 4601 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2120breq2d 5122 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
2220breq1d 5120 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
2321, 22anbi12d 632 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
2423rexrab 3659 . . . . . . . . . . . . 13 (∃𝑦 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ( bday 𝑦) = 𝑧 ↔ ∃𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ ( bday 𝑦) = 𝑧))
2519, 24bitri 275 . . . . . . . . . . . 12 (𝑧 ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∃𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ ( bday 𝑦) = 𝑧))
2625imbi1i 350 . . . . . . . . . . 11 ((𝑧 ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → ( bday 𝑋) ⊆ 𝑧) ↔ (∃𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ ( bday 𝑦) = 𝑧) → ( bday 𝑋) ⊆ 𝑧))
27 r19.23v 3180 . . . . . . . . . . 11 (∀𝑦 No (((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ ( bday 𝑦) = 𝑧) → ( bday 𝑋) ⊆ 𝑧) ↔ (∃𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ ( bday 𝑦) = 𝑧) → ( bday 𝑋) ⊆ 𝑧))
28 eqcom 2744 . . . . . . . . . . . . . . 15 (( bday 𝑦) = 𝑧𝑧 = ( bday 𝑦))
2928anbi1ci 627 . . . . . . . . . . . . . 14 (((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ ( bday 𝑦) = 𝑧) ↔ (𝑧 = ( bday 𝑦) ∧ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
3029imbi1i 350 . . . . . . . . . . . . 13 ((((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ ( bday 𝑦) = 𝑧) → ( bday 𝑋) ⊆ 𝑧) ↔ ((𝑧 = ( bday 𝑦) ∧ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)) → ( bday 𝑋) ⊆ 𝑧))
31 impexp 452 . . . . . . . . . . . . 13 (((𝑧 = ( bday 𝑦) ∧ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)) → ( bday 𝑋) ⊆ 𝑧) ↔ (𝑧 = ( bday 𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ 𝑧)))
3230, 31bitri 275 . . . . . . . . . . . 12 ((((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ ( bday 𝑦) = 𝑧) → ( bday 𝑋) ⊆ 𝑧) ↔ (𝑧 = ( bday 𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ 𝑧)))
3332ralbii 3097 . . . . . . . . . . 11 (∀𝑦 No (((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ ( bday 𝑦) = 𝑧) → ( bday 𝑋) ⊆ 𝑧) ↔ ∀𝑦 No (𝑧 = ( bday 𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ 𝑧)))
3426, 27, 333bitr2i 299 . . . . . . . . . 10 ((𝑧 ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → ( bday 𝑋) ⊆ 𝑧) ↔ ∀𝑦 No (𝑧 = ( bday 𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ 𝑧)))
3534albii 1822 . . . . . . . . 9 (∀𝑧(𝑧 ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → ( bday 𝑋) ⊆ 𝑧) ↔ ∀𝑧𝑦 No (𝑧 = ( bday 𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ 𝑧)))
36 df-ral 3066 . . . . . . . . 9 (∀𝑧 ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})( bday 𝑋) ⊆ 𝑧 ↔ ∀𝑧(𝑧 ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → ( bday 𝑋) ⊆ 𝑧))
37 ralcom4 3272 . . . . . . . . 9 (∀𝑦 No 𝑧(𝑧 = ( bday 𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ 𝑧)) ↔ ∀𝑧𝑦 No (𝑧 = ( bday 𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ 𝑧)))
3835, 36, 373bitr4i 303 . . . . . . . 8 (∀𝑧 ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})( bday 𝑋) ⊆ 𝑧 ↔ ∀𝑦 No 𝑧(𝑧 = ( bday 𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ 𝑧)))
39 fvex 6860 . . . . . . . . . 10 ( bday 𝑦) ∈ V
40 sseq2 3975 . . . . . . . . . . 11 (𝑧 = ( bday 𝑦) → (( bday 𝑋) ⊆ 𝑧 ↔ ( bday 𝑋) ⊆ ( bday 𝑦)))
4140imbi2d 341 . . . . . . . . . 10 (𝑧 = ( bday 𝑦) → (((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ 𝑧) ↔ ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦))))
4239, 41ceqsalv 3484 . . . . . . . . 9 (∀𝑧(𝑧 = ( bday 𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ 𝑧)) ↔ ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦)))
4342ralbii 3097 . . . . . . . 8 (∀𝑦 No 𝑧(𝑧 = ( bday 𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ 𝑧)) ↔ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦)))
4438, 43bitri 275 . . . . . . 7 (∀𝑧 ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})( bday 𝑋) ⊆ 𝑧 ↔ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦)))
4517, 44bitri 275 . . . . . 6 (( bday 𝑋) ⊆ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦)))
4616, 45bitr3di 286 . . . . 5 (((𝐿 <<s 𝑅𝑋 No ) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → ((( bday 𝑋) ⊆ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ∧ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ ( bday 𝑋)) ↔ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦))))
472, 46bitrid 283 . . . 4 (((𝐿 <<s 𝑅𝑋 No ) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → (( bday 𝑋) = ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦))))
4847pm5.32da 580 . . 3 ((𝐿 <<s 𝑅𝑋 No ) → (((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅) ∧ ( bday 𝑋) = ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})) ↔ ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅) ∧ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦)))))
49 df-3an 1090 . . 3 ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday 𝑋) = ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})) ↔ ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅) ∧ ( bday 𝑋) = ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})))
50 df-3an 1090 . . 3 ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦))) ↔ ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅) ∧ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦))))
5148, 49, 503bitr4g 314 . 2 ((𝐿 <<s 𝑅𝑋 No ) → ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday 𝑋) = ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})) ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦)))))
521, 51bitrd 279 1 ((𝐿 <<s 𝑅𝑋 No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wcel 2107  wral 3065  wrex 3074  {crab 3410  wss 3915  {csn 4591   cint 4912   class class class wbr 5110  cima 5641   Fn wfn 6496  cfv 6501  (class class class)co 7362   No csur 27004   bday cbday 27006   <<s csslt 27142   |s cscut 27144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1o 8417  df-2o 8418  df-no 27007  df-slt 27008  df-bday 27009  df-sslt 27143  df-scut 27145
This theorem is referenced by:  bday0b  27191
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