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Theorem scutun12 32042
Description: Union law for surreal cuts. (Contributed by Scott Fenton, 9-Dec-2021.)
Assertion
Ref Expression
scutun12 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴𝐶) |s (𝐵𝐷)) = (𝐴 |s 𝐵))

Proof of Theorem scutun12
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1081 . . . . . . 7 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s 𝐵)
2 scutcut 32037 . . . . . . 7 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
31, 2syl 17 . . . . . 6 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
43simp2d 1094 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
5 simp2 1082 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐶 <<s {(𝐴 |s 𝐵)})
6 ssltun1 32040 . . . . 5 ((𝐴 <<s {(𝐴 |s 𝐵)} ∧ 𝐶 <<s {(𝐴 |s 𝐵)}) → (𝐴𝐶) <<s {(𝐴 |s 𝐵)})
74, 5, 6syl2anc 694 . . . 4 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴𝐶) <<s {(𝐴 |s 𝐵)})
83simp3d 1095 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐵)
9 simp3 1083 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐷)
10 ssltun2 32041 . . . . 5 (({(𝐴 |s 𝐵)} <<s 𝐵 ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵𝐷))
118, 9, 10syl2anc 694 . . . 4 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵𝐷))
12 ovex 6718 . . . . . 6 (𝐴 |s 𝐵) ∈ V
1312snnz 4340 . . . . 5 {(𝐴 |s 𝐵)} ≠ ∅
14 sslttr 32039 . . . . 5 (((𝐴𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵𝐷) ∧ {(𝐴 |s 𝐵)} ≠ ∅) → (𝐴𝐶) <<s (𝐵𝐷))
1513, 14mp3an3 1453 . . . 4 (((𝐴𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵𝐷)) → (𝐴𝐶) <<s (𝐵𝐷))
167, 11, 15syl2anc 694 . . 3 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴𝐶) <<s (𝐵𝐷))
17 scutval 32036 . . 3 ((𝐴𝐶) <<s (𝐵𝐷) → ((𝐴𝐶) |s (𝐵𝐷)) = (𝑥 ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ( bday 𝑥) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))})))
1816, 17syl 17 . 2 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴𝐶) |s (𝐵𝐷)) = (𝑥 ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ( bday 𝑥) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))})))
19 vex 3234 . . . . . . . . . 10 𝑥 ∈ V
2019elima 5506 . . . . . . . . 9 (𝑥 ∈ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) ↔ ∃𝑧 ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}𝑧 bday 𝑥)
21 sneq 4220 . . . . . . . . . . . 12 (𝑦 = 𝑧 → {𝑦} = {𝑧})
2221breq2d 4697 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝐴𝐶) <<s {𝑦} ↔ (𝐴𝐶) <<s {𝑧}))
2321breq1d 4695 . . . . . . . . . . 11 (𝑦 = 𝑧 → ({𝑦} <<s (𝐵𝐷) ↔ {𝑧} <<s (𝐵𝐷)))
2422, 23anbi12d 747 . . . . . . . . . 10 (𝑦 = 𝑧 → (((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷)) ↔ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))))
2524rexrab 3403 . . . . . . . . 9 (∃𝑧 ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}𝑧 bday 𝑥 ↔ ∃𝑧 No (((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷)) ∧ 𝑧 bday 𝑥))
2620, 25bitri 264 . . . . . . . 8 (𝑥 ∈ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) ↔ ∃𝑧 No (((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷)) ∧ 𝑧 bday 𝑥))
27 simplr 807 . . . . . . . . . . . . 13 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → 𝑧 No )
28 bdayfn 32014 . . . . . . . . . . . . . 14 bday Fn No
29 fnbrfvb 6274 . . . . . . . . . . . . . 14 (( bday Fn No 𝑧 No ) → (( bday 𝑧) = 𝑥𝑧 bday 𝑥))
3028, 29mpan 706 . . . . . . . . . . . . 13 (𝑧 No → (( bday 𝑧) = 𝑥𝑧 bday 𝑥))
3127, 30syl 17 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → (( bday 𝑧) = 𝑥𝑧 bday 𝑥))
32 simpll1 1120 . . . . . . . . . . . . . . 15 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → 𝐴 <<s 𝐵)
33 scutbday 32038 . . . . . . . . . . . . . . 15 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
3432, 33syl 17 . . . . . . . . . . . . . 14 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
35 simprl 809 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → (𝐴𝐶) <<s {𝑧})
36 ssun1 3809 . . . . . . . . . . . . . . . . . . . 20 𝐴 ⊆ (𝐴𝐶)
37 sssslt1 32031 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝐶) <<s {𝑧} ∧ 𝐴 ⊆ (𝐴𝐶)) → 𝐴 <<s {𝑧})
3836, 37mpan2 707 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝐶) <<s {𝑧} → 𝐴 <<s {𝑧})
3935, 38syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → 𝐴 <<s {𝑧})
40 simprr 811 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → {𝑧} <<s (𝐵𝐷))
41 ssun1 3809 . . . . . . . . . . . . . . . . . . . 20 𝐵 ⊆ (𝐵𝐷)
42 sssslt2 32032 . . . . . . . . . . . . . . . . . . . 20 (({𝑧} <<s (𝐵𝐷) ∧ 𝐵 ⊆ (𝐵𝐷)) → {𝑧} <<s 𝐵)
4341, 42mpan2 707 . . . . . . . . . . . . . . . . . . 19 ({𝑧} <<s (𝐵𝐷) → {𝑧} <<s 𝐵)
4440, 43syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → {𝑧} <<s 𝐵)
4539, 44jca 553 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵))
4621breq2d 4697 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑧}))
4721breq1d 4695 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → ({𝑦} <<s 𝐵 ↔ {𝑧} <<s 𝐵))
4846, 47anbi12d 747 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
4948elrab 3396 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑧 No ∧ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
5027, 45, 49sylanbrc 699 . . . . . . . . . . . . . . . 16 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → 𝑧 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
51 ssrab2 3720 . . . . . . . . . . . . . . . . 17 {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
52 fnfvima 6536 . . . . . . . . . . . . . . . . 17 (( bday Fn No ∧ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No 𝑧 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ( bday 𝑧) ∈ ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
5328, 51, 52mp3an12 1454 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} → ( bday 𝑧) ∈ ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
5450, 53syl 17 . . . . . . . . . . . . . . 15 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → ( bday 𝑧) ∈ ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
55 intss1 4524 . . . . . . . . . . . . . . 15 (( bday 𝑧) ∈ ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday 𝑧))
5654, 55syl 17 . . . . . . . . . . . . . 14 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday 𝑧))
5734, 56eqsstrd 3672 . . . . . . . . . . . . 13 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑧))
58 sseq2 3660 . . . . . . . . . . . . . . 15 (( bday 𝑧) = 𝑥 → (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑧) ↔ ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
5958biimpd 219 . . . . . . . . . . . . . 14 (( bday 𝑧) = 𝑥 → (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑧) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
6059com12 32 . . . . . . . . . . . . 13 (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑧) → (( bday 𝑧) = 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
6157, 60syl 17 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → (( bday 𝑧) = 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
6231, 61sylbird 250 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) ∧ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))) → (𝑧 bday 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
6362ex 449 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) → (((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷)) → (𝑧 bday 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥)))
6463impd 446 . . . . . . . . 9 (((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 No ) → ((((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷)) ∧ 𝑧 bday 𝑥) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
6564rexlimdva 3060 . . . . . . . 8 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (∃𝑧 No (((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷)) ∧ 𝑧 bday 𝑥) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
6626, 65syl5bi 232 . . . . . . 7 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝑥 ∈ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
6766ralrimiv 2994 . . . . . 6 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ∀𝑥 ∈ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))})( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥)
68 ssint 4525 . . . . . 6 (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) ↔ ∀𝑥 ∈ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))})( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥)
6967, 68sylibr 224 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}))
703simp1d 1093 . . . . . . . 8 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) ∈ No )
717, 11jca 553 . . . . . . . 8 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵𝐷)))
72 sneq 4220 . . . . . . . . . . 11 (𝑦 = (𝐴 |s 𝐵) → {𝑦} = {(𝐴 |s 𝐵)})
7372breq2d 4697 . . . . . . . . . 10 (𝑦 = (𝐴 |s 𝐵) → ((𝐴𝐶) <<s {𝑦} ↔ (𝐴𝐶) <<s {(𝐴 |s 𝐵)}))
7472breq1d 4695 . . . . . . . . . 10 (𝑦 = (𝐴 |s 𝐵) → ({𝑦} <<s (𝐵𝐷) ↔ {(𝐴 |s 𝐵)} <<s (𝐵𝐷)))
7573, 74anbi12d 747 . . . . . . . . 9 (𝑦 = (𝐴 |s 𝐵) → (((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷)) ↔ ((𝐴𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵𝐷))))
7675elrab 3396 . . . . . . . 8 ((𝐴 |s 𝐵) ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ↔ ((𝐴 |s 𝐵) ∈ No ∧ ((𝐴𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵𝐷))))
7770, 71, 76sylanbrc 699 . . . . . . 7 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))})
78 ssrab2 3720 . . . . . . . 8 {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ⊆ No
79 fnfvima 6536 . . . . . . . 8 (( bday Fn No ∧ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ⊆ No ∧ (𝐴 |s 𝐵) ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}))
8028, 78, 79mp3an12 1454 . . . . . . 7 ((𝐴 |s 𝐵) ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}))
8177, 80syl 17 . . . . . 6 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}))
82 intss1 4524 . . . . . 6 (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) → ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) ⊆ ( bday ‘(𝐴 |s 𝐵)))
8381, 82syl 17 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) ⊆ ( bday ‘(𝐴 |s 𝐵)))
8469, 83eqssd 3653 . . . 4 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}))
85 conway 32035 . . . . . 6 ((𝐴𝐶) <<s (𝐵𝐷) → ∃!𝑥 ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ( bday 𝑥) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}))
8616, 85syl 17 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ∃!𝑥 ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ( bday 𝑥) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}))
87 fveq2 6229 . . . . . . 7 (𝑥 = (𝐴 |s 𝐵) → ( bday 𝑥) = ( bday ‘(𝐴 |s 𝐵)))
8887eqeq1d 2653 . . . . . 6 (𝑥 = (𝐴 |s 𝐵) → (( bday 𝑥) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) ↔ ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))})))
8988riota2 6673 . . . . 5 (((𝐴 |s 𝐵) ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ∧ ∃!𝑥 ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ( bday 𝑥) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))})) → (( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) ↔ (𝑥 ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ( bday 𝑥) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))})) = (𝐴 |s 𝐵)))
9077, 86, 89syl2anc 694 . . . 4 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) ↔ (𝑥 ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ( bday 𝑥) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))})) = (𝐴 |s 𝐵)))
9184, 90mpbid 222 . . 3 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝑥 ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ( bday 𝑥) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))})) = (𝐴 |s 𝐵))
9291eqcomd 2657 . 2 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) = (𝑥 ∈ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ( bday 𝑥) = ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))})))
9318, 92eqtr4d 2688 1 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴𝐶) |s (𝐵𝐷)) = (𝐴 |s 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  ∃!wreu 2943  {crab 2945  cun 3605  wss 3607  c0 3948  {csn 4210   cint 4507   class class class wbr 4685  cima 5146   Fn wfn 5921  cfv 5926  crio 6650  (class class class)co 6690   No csur 31918   bday cbday 31920   <<s csslt 32021   |s cscut 32023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1o 7605  df-2o 7606  df-no 31921  df-slt 31922  df-bday 31923  df-sslt 32022  df-scut 32024
This theorem is referenced by: (None)
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