MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqscut2 Structured version   Visualization version   GIF version

Theorem eqscut2 27689
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 27688 . 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 3992 . . . . 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 4633 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → {𝑥} = {𝑋})
43breq2d 5153 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
53breq1d 5151 . . . . . . . . . . . 12 (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
64, 5anbi12d 630 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
76elrab3 3679 . . . . . . . . . 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 27656 . . . . . . . . 9 bday Fn No
11 ssrab2 4072 . . . . . . . . 9 {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
12 fnfvima 7229 . . . . . . . . 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 1447 . . . . . . . 8 (𝑋 ∈ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} → ( bday 𝑋) ∈ ( bday “ {𝑥 No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
14 intss1 4960 . . . . . . . 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 4961 . . . . . . 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 6957 . . . . . . . . . . . . . 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 4633 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2120breq2d 5153 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
2220breq1d 5151 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
2321, 22anbi12d 630 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
2423rexrab 3687 . . . . . . . . . . . . 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 3176 . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . 15 (( bday 𝑦) = 𝑧𝑧 = ( bday 𝑦))
2928anbi1ci 625 . . . . . . . . . . . . . 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 3087 . . . . . . . . . . 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 1813 . . . . . . . . 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 3056 . . . . . . . . 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 3277 . . . . . . . . 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 6897 . . . . . . . . . 10 ( bday 𝑦) ∈ V
40 sseq2 4003 . . . . . . . . . . 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 3506 . . . . . . . . 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 3087 . . . . . . . 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 578 . . 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 1086 . . 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 1086 . . 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 395  w3a 1084  wal 1531   = wceq 1533  wcel 2098  wral 3055  wrex 3064  {crab 3426  wss 3943  {csn 4623   cint 4943   class class class wbr 5141  cima 5672   Fn wfn 6531  cfv 6536  (class class class)co 7404   No csur 27523   bday cbday 27525   <<s csslt 27663   |s cscut 27665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1o 8464  df-2o 8465  df-no 27526  df-slt 27527  df-bday 27528  df-sslt 27664  df-scut 27666
This theorem is referenced by:  bday0b  27713  cuteq1  27716
  Copyright terms: Public domain W3C validator