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Theorem scutun12 27730
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 1134 . . . . . . 7 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s 𝐵)
2 scutcut 27721 . . . . . . 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 1141 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
5 simp2 1135 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐶 <<s {(𝐴 |s 𝐵)})
6 ssltun1 27728 . . . . 5 ((𝐴 <<s {(𝐴 |s 𝐵)} ∧ 𝐶 <<s {(𝐴 |s 𝐵)}) → (𝐴𝐶) <<s {(𝐴 |s 𝐵)})
74, 5, 6syl2anc 583 . . . 4 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴𝐶) <<s {(𝐴 |s 𝐵)})
83simp3d 1142 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐵)
9 simp3 1136 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐷)
10 ssltun2 27729 . . . . 5 (({(𝐴 |s 𝐵)} <<s 𝐵 ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵𝐷))
118, 9, 10syl2anc 583 . . . 4 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵𝐷))
12 ovex 7447 . . . . . 6 (𝐴 |s 𝐵) ∈ V
1312snnz 4776 . . . . 5 {(𝐴 |s 𝐵)} ≠ ∅
14 sslttr 27727 . . . . 5 (((𝐴𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵𝐷) ∧ {(𝐴 |s 𝐵)} ≠ ∅) → (𝐴𝐶) <<s (𝐵𝐷))
1513, 14mp3an3 1447 . . . 4 (((𝐴𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵𝐷)) → (𝐴𝐶) <<s (𝐵𝐷))
167, 11, 15syl2anc 583 . . 3 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴𝐶) <<s (𝐵𝐷))
17 scutval 27720 . . 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 3473 . . . . . . . . . 10 𝑥 ∈ V
2019elima 6062 . . . . . . . . 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 4634 . . . . . . . . . . . 12 (𝑦 = 𝑧 → {𝑦} = {𝑧})
2221breq2d 5154 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝐴𝐶) <<s {𝑦} ↔ (𝐴𝐶) <<s {𝑧}))
2321breq1d 5152 . . . . . . . . . . 11 (𝑦 = 𝑧 → ({𝑦} <<s (𝐵𝐷) ↔ {𝑧} <<s (𝐵𝐷)))
2422, 23anbi12d 630 . . . . . . . . . 10 (𝑦 = 𝑧 → (((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷)) ↔ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))))
2524rexrab 3689 . . . . . . . . 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 275 . . . . . . . 8 (𝑥 ∈ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) ↔ ∃𝑧 No (((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷)) ∧ 𝑧 bday 𝑥))
27 simplr 768 . . . . . . . . . . . . 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 27693 . . . . . . . . . . . . . 14 bday Fn No
29 fnbrfvb 6944 . . . . . . . . . . . . . 14 (( bday Fn No 𝑧 No ) → (( bday 𝑧) = 𝑥𝑧 bday 𝑥))
3028, 29mpan 689 . . . . . . . . . . . . 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 1210 . . . . . . . . . . . . . . 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 27724 . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . 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 4168 . . . . . . . . . . . . . . . . . . . 20 𝐴 ⊆ (𝐴𝐶)
37 sssslt1 27715 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝐶) <<s {𝑧} ∧ 𝐴 ⊆ (𝐴𝐶)) → 𝐴 <<s {𝑧})
3836, 37mpan2 690 . . . . . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . . . . 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 4168 . . . . . . . . . . . . . . . . . . . 20 𝐵 ⊆ (𝐵𝐷)
42 sssslt2 27716 . . . . . . . . . . . . . . . . . . . 20 (({𝑧} <<s (𝐵𝐷) ∧ 𝐵 ⊆ (𝐵𝐷)) → {𝑧} <<s 𝐵)
4341, 42mpan2 690 . . . . . . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . . . . . 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 5154 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑧}))
4721breq1d 5152 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → ({𝑦} <<s 𝐵 ↔ {𝑧} <<s 𝐵))
4846, 47anbi12d 630 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
4948elrab 3680 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑧 No ∧ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
5027, 45, 49sylanbrc 582 . . . . . . . . . . . . . . . 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 4073 . . . . . . . . . . . . . . . . 17 {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
52 fnfvima 7239 . . . . . . . . . . . . . . . . 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 1448 . . . . . . . . . . . . . . . 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 4961 . . . . . . . . . . . . . . 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 4016 . . . . . . . . . . . . 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 4004 . . . . . . . . . . . . . . 15 (( bday 𝑧) = 𝑥 → (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑧) ↔ ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
5958biimpd 228 . . . . . . . . . . . . . 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 260 . . . . . . . . . . 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 412 . . . . . . . . . 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 410 . . . . . . . . 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 3150 . . . . . . . 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, 65biimtrid 241 . . . . . . 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 3140 . . . . . 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 4962 . . . . . 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 233 . . . . 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 1140 . . . . . . . 8 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) ∈ No )
717, 11jca 511 . . . . . . . 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 4634 . . . . . . . . . . 11 (𝑦 = (𝐴 |s 𝐵) → {𝑦} = {(𝐴 |s 𝐵)})
7372breq2d 5154 . . . . . . . . . 10 (𝑦 = (𝐴 |s 𝐵) → ((𝐴𝐶) <<s {𝑦} ↔ (𝐴𝐶) <<s {(𝐴 |s 𝐵)}))
7472breq1d 5152 . . . . . . . . . 10 (𝑦 = (𝐴 |s 𝐵) → ({𝑦} <<s (𝐵𝐷) ↔ {(𝐴 |s 𝐵)} <<s (𝐵𝐷)))
7573, 74anbi12d 630 . . . . . . . . 9 (𝑦 = (𝐴 |s 𝐵) → (((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷)) ↔ ((𝐴𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵𝐷))))
7675elrab 3680 . . . . . . . 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 582 . . . . . . 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 4073 . . . . . . . 8 {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ⊆ No
79 fnfvima 7239 . . . . . . . 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 1448 . . . . . . 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 4961 . . . . . 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 3995 . . . 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 27719 . . . . . 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 fveqeq2 6900 . . . . . 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))})))
8887riota2 7396 . . . . 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)))
8977, 86, 88syl2anc 583 . . . 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)))
9084, 89mpbid 231 . . 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))
9190eqcomd 2733 . 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))})))
9218, 91eqtr4d 2770 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 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2935  wral 3056  wrex 3065  ∃!wreu 3369  {crab 3427  cun 3942  wss 3944  c0 4318  {csn 4624   cint 4944   class class class wbr 5142  cima 5675   Fn wfn 6537  cfv 6542  crio 7369  (class class class)co 7414   No csur 27560   bday cbday 27562   <<s csslt 27700   |s cscut 27702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1o 8480  df-2o 8481  df-no 27563  df-slt 27564  df-bday 27565  df-sslt 27701  df-scut 27703
This theorem is referenced by: (None)
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