Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqscut2 Structured version   Visualization version   GIF version

Theorem eqscut2 34091
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 34090 . 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 3946 . . . . 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 4582 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → {𝑥} = {𝑋})
43breq2d 5101 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
53breq1d 5099 . . . . . . . . . . . 12 (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
64, 5anbi12d 631 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
76elrab3 3635 . . . . . . . . . 10 (𝑋 No → (𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
87adantl 482 . . . . . . . . 9 ((𝐿 <<s 𝑅𝑋 No ) → (𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
98biimpar 478 . . . . . . . 8 (((𝐿 <<s 𝑅𝑋 No ) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → 𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})
10 bdayfn 34059 . . . . . . . . 9 bday Fn No
11 ssrab2 4024 . . . . . . . . 9 {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
12 fnfvima 7159 . . . . . . . . 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 1450 . . . . . . . 8 (𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} → ( bday 𝑋) ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
14 intss1 4908 . . . . . . . 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 532 . . . . . 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 4909 . . . . . . 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 6891 . . . . . . . . . . . . . 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 689 . . . . . . . . . . . . 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 4582 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2120breq2d 5101 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
2220breq1d 5099 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
2321, 22anbi12d 631 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
2423rexrab 3643 . . . . . . . . . . . . 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 274 . . . . . . . . . . . 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 3175 . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . 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 274 . . . . . . . . . . . 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 3092 . . . . . . . . . . 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 298 . . . . . . . . . 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 1820 . . . . . . . . 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 3265 . . . . . . . . 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 302 . . . . . . . 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 6832 . . . . . . . . . 10 ( bday 𝑦) ∈ V
40 sseq2 3957 . . . . . . . . . . 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 3476 . . . . . . . . 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 3092 . . . . . . . 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 274 . . . . . . 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 274 . . . . . 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 285 . . . . 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 282 . . . 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 1088 . . 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 1088 . . 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 313 . 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 278 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 396  w3a 1086  wal 1538   = wceq 1540  wcel 2105  wral 3061  wrex 3070  {crab 3403  wss 3897  {csn 4572   cint 4893   class class class wbr 5089  cima 5617   Fn wfn 6468  cfv 6473  (class class class)co 7329   No csur 26886   bday cbday 26888   <<s csslt 34066   |s cscut 34068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6299  df-on 6300  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-1o 8359  df-2o 8360  df-no 26889  df-slt 26890  df-bday 26891  df-sslt 34067  df-scut 34069
This theorem is referenced by:  bday0b  34115
  Copyright terms: Public domain W3C validator