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Theorem scutun12 27149
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 1136 . . . . . . 7 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s 𝐵)
2 scutcut 27140 . . . . . . 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 1143 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
5 simp2 1137 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐶 <<s {(𝐴 |s 𝐵)})
6 ssltun1 27147 . . . . 5 ((𝐴 <<s {(𝐴 |s 𝐵)} ∧ 𝐶 <<s {(𝐴 |s 𝐵)}) → (𝐴𝐶) <<s {(𝐴 |s 𝐵)})
74, 5, 6syl2anc 584 . . . 4 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴𝐶) <<s {(𝐴 |s 𝐵)})
83simp3d 1144 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐵)
9 simp3 1138 . . . . 5 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐷)
10 ssltun2 27148 . . . . 5 (({(𝐴 |s 𝐵)} <<s 𝐵 ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵𝐷))
118, 9, 10syl2anc 584 . . . 4 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵𝐷))
12 ovex 7390 . . . . . 6 (𝐴 |s 𝐵) ∈ V
1312snnz 4737 . . . . 5 {(𝐴 |s 𝐵)} ≠ ∅
14 sslttr 27146 . . . . 5 (((𝐴𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵𝐷) ∧ {(𝐴 |s 𝐵)} ≠ ∅) → (𝐴𝐶) <<s (𝐵𝐷))
1513, 14mp3an3 1450 . . . 4 (((𝐴𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵𝐷)) → (𝐴𝐶) <<s (𝐵𝐷))
167, 11, 15syl2anc 584 . . 3 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴𝐶) <<s (𝐵𝐷))
17 scutval 27139 . . 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 3449 . . . . . . . . . 10 𝑥 ∈ V
2019elima 6018 . . . . . . . . 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 4596 . . . . . . . . . . . 12 (𝑦 = 𝑧 → {𝑦} = {𝑧})
2221breq2d 5117 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝐴𝐶) <<s {𝑦} ↔ (𝐴𝐶) <<s {𝑧}))
2321breq1d 5115 . . . . . . . . . . 11 (𝑦 = 𝑧 → ({𝑦} <<s (𝐵𝐷) ↔ {𝑧} <<s (𝐵𝐷)))
2422, 23anbi12d 631 . . . . . . . . . 10 (𝑦 = 𝑧 → (((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷)) ↔ ((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷))))
2524rexrab 3654 . . . . . . . . 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 274 . . . . . . . 8 (𝑥 ∈ ( bday “ {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))}) ↔ ∃𝑧 No (((𝐴𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵𝐷)) ∧ 𝑧 bday 𝑥))
27 simplr 767 . . . . . . . . . . . . 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 27113 . . . . . . . . . . . . . 14 bday Fn No
29 fnbrfvb 6895 . . . . . . . . . . . . . 14 (( bday Fn No 𝑧 No ) → (( bday 𝑧) = 𝑥𝑧 bday 𝑥))
3028, 29mpan 688 . . . . . . . . . . . . 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 1212 . . . . . . . . . . . . . . 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 27143 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . 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 4132 . . . . . . . . . . . . . . . . . . . 20 𝐴 ⊆ (𝐴𝐶)
37 sssslt1 27134 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝐶) <<s {𝑧} ∧ 𝐴 ⊆ (𝐴𝐶)) → 𝐴 <<s {𝑧})
3836, 37mpan2 689 . . . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . . 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 4132 . . . . . . . . . . . . . . . . . . . 20 𝐵 ⊆ (𝐵𝐷)
42 sssslt2 27135 . . . . . . . . . . . . . . . . . . . 20 (({𝑧} <<s (𝐵𝐷) ∧ 𝐵 ⊆ (𝐵𝐷)) → {𝑧} <<s 𝐵)
4341, 42mpan2 689 . . . . . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . . . . . 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 5117 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑧}))
4721breq1d 5115 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → ({𝑦} <<s 𝐵 ↔ {𝑧} <<s 𝐵))
4846, 47anbi12d 631 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
4948elrab 3645 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑧 No ∧ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
5027, 45, 49sylanbrc 583 . . . . . . . . . . . . . . . 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 4037 . . . . . . . . . . . . . . . . 17 {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
52 fnfvima 7183 . . . . . . . . . . . . . . . . 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 1451 . . . . . . . . . . . . . . . 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 4924 . . . . . . . . . . . . . . 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 3982 . . . . . . . . . . . . 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 3970 . . . . . . . . . . . . . . 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 259 . . . . . . . . . . 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 413 . . . . . . . . . 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 411 . . . . . . . . 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 3152 . . . . . . . 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 3142 . . . . . 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 4925 . . . . . 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 1142 . . . . . . . 8 ((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) ∈ No )
717, 11jca 512 . . . . . . . 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 4596 . . . . . . . . . . 11 (𝑦 = (𝐴 |s 𝐵) → {𝑦} = {(𝐴 |s 𝐵)})
7372breq2d 5117 . . . . . . . . . 10 (𝑦 = (𝐴 |s 𝐵) → ((𝐴𝐶) <<s {𝑦} ↔ (𝐴𝐶) <<s {(𝐴 |s 𝐵)}))
7472breq1d 5115 . . . . . . . . . 10 (𝑦 = (𝐴 |s 𝐵) → ({𝑦} <<s (𝐵𝐷) ↔ {(𝐴 |s 𝐵)} <<s (𝐵𝐷)))
7573, 74anbi12d 631 . . . . . . . . 9 (𝑦 = (𝐴 |s 𝐵) → (((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷)) ↔ ((𝐴𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵𝐷))))
7675elrab 3645 . . . . . . . 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 583 . . . . . . 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 4037 . . . . . . . 8 {𝑦 No ∣ ((𝐴𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵𝐷))} ⊆ No
79 fnfvima 7183 . . . . . . . 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 1451 . . . . . . 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 4924 . . . . . 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 3961 . . . 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 27138 . . . . . 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 6851 . . . . . 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 7339 . . . . 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 584 . . . 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 2742 . 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 2779 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 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  ∃!wreu 3351  {crab 3407  cun 3908  wss 3910  c0 4282  {csn 4586   cint 4907   class class class wbr 5105  cima 5636   Fn wfn 6491  cfv 6496  crio 7312  (class class class)co 7357   No csur 26988   bday cbday 26990   <<s csslt 27120   |s cscut 27122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1o 8412  df-2o 8413  df-no 26991  df-slt 26992  df-bday 26993  df-sslt 27121  df-scut 27123
This theorem is referenced by: (None)
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