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Theorem eqscut2 27851
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 27850 . 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 3999 . . . . 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 4636 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → {𝑥} = {𝑋})
43breq2d 5155 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
53breq1d 5153 . . . . . . . . . . . 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 3693 . . . . . . . . . 10 (𝑋 No → (𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
87adantl 481 . . . . . . . . 9 ((𝐿 <<s 𝑅𝑋 No ) → (𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
98biimpar 477 . . . . . . . 8 (((𝐿 <<s 𝑅𝑋 No ) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → 𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})
10 bdayfn 27818 . . . . . . . . 9 bday Fn No
11 ssrab2 4080 . . . . . . . . 9 {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
12 fnfvima 7253 . . . . . . . . 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 1453 . . . . . . . 8 (𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} → ( bday 𝑋) ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
14 intss1 4963 . . . . . . . 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 531 . . . . . 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 4964 . . . . . . 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 6981 . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . 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 4636 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2120breq2d 5155 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
2220breq1d 5153 . . . . . . . . . . . . . . 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 3702 . . . . . . . . . . . . 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 349 . . . . . . . . . . 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 3183 . . . . . . . . . . 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 626 . . . . . . . . . . . . . 14 (((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ ( bday 𝑦) = 𝑧) ↔ (𝑧 = ( bday 𝑦) ∧ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
3029imbi1i 349 . . . . . . . . . . . . 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 450 . . . . . . . . . . . . 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 3093 . . . . . . . . . . 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 1819 . . . . . . . . 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 3062 . . . . . . . . 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 3286 . . . . . . . . 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 6919 . . . . . . . . . 10 ( bday 𝑦) ∈ V
40 sseq2 4010 . . . . . . . . . . 11 (𝑧 = ( bday 𝑦) → (( bday 𝑋) ⊆ 𝑧 ↔ ( bday 𝑋) ⊆ ( bday 𝑦)))
4140imbi2d 340 . . . . . . . . . 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 3521 . . . . . . . . 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 3093 . . . . . . . 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 579 . . 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 1089 . . 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 1089 . . 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 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  wss 3951  {csn 4626   cint 4946   class class class wbr 5143  cima 5688   Fn wfn 6556  cfv 6561  (class class class)co 7431   No csur 27684   bday cbday 27686   <<s csslt 27825   |s cscut 27827
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828
This theorem is referenced by:  bday0b  27875  cuteq1  27878
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