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Theorem dfsalgen2 45057
Description: Alternate characterization of the sigma-algebra generated by a set. It is the smallest sigma-algebra, on the same base set, that includes the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypothesis
Ref Expression
dfsalgen2.1 (πœ‘ β†’ 𝑋 ∈ 𝑉)
Assertion
Ref Expression
dfsalgen2 (πœ‘ β†’ ((SalGenβ€˜π‘‹) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))))
Distinct variable groups:   𝑦,𝑆   𝑦,𝑋   πœ‘,𝑦
Allowed substitution hint:   𝑉(𝑦)
Proof of Theorem dfsalgen2
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . . 8 ((SalGenβ€˜π‘‹) = 𝑆 β†’ (SalGenβ€˜π‘‹) = 𝑆)
21eqcomd 2739 . . . . . . 7 ((SalGenβ€˜π‘‹) = 𝑆 β†’ 𝑆 = (SalGenβ€˜π‘‹))
32adantl 483 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ 𝑆 = (SalGenβ€˜π‘‹))
4 dfsalgen2.1 . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ 𝑉)
5 salgencl 45048 . . . . . . . 8 (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
64, 5syl 17 . . . . . . 7 (πœ‘ β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
76adantr 482 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
83, 7eqeltrd 2834 . . . . 5 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ 𝑆 ∈ SAlg)
9 unieq 4920 . . . . . . 7 ((SalGenβ€˜π‘‹) = 𝑆 β†’ βˆͺ (SalGenβ€˜π‘‹) = βˆͺ 𝑆)
109adantl 483 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ βˆͺ (SalGenβ€˜π‘‹) = βˆͺ 𝑆)
114adantr 482 . . . . . . 7 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ 𝑋 ∈ 𝑉)
12 eqid 2733 . . . . . . 7 (SalGenβ€˜π‘‹) = (SalGenβ€˜π‘‹)
13 eqid 2733 . . . . . . 7 βˆͺ 𝑋 = βˆͺ 𝑋
1411, 12, 13salgenuni 45053 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ βˆͺ (SalGenβ€˜π‘‹) = βˆͺ 𝑋)
1510, 14eqtr3d 2775 . . . . 5 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ βˆͺ 𝑆 = βˆͺ 𝑋)
1612sssalgen 45051 . . . . . . 7 (𝑋 ∈ 𝑉 β†’ 𝑋 βŠ† (SalGenβ€˜π‘‹))
1711, 16syl 17 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ 𝑋 βŠ† (SalGenβ€˜π‘‹))
18 simpr 486 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ (SalGenβ€˜π‘‹) = 𝑆)
1917, 18sseqtrd 4023 . . . . 5 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ 𝑋 βŠ† 𝑆)
208, 15, 193jca 1129 . . . 4 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ (𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆))
213ad2antrr 725 . . . . . . . 8 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 = (SalGenβ€˜π‘‹))
2221adantrl 715 . . . . . . 7 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑆 = (SalGenβ€˜π‘‹))
2311ad2antrr 725 . . . . . . . . 9 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 βŠ† 𝑦) β†’ 𝑋 ∈ 𝑉)
2423adantrl 715 . . . . . . . 8 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑋 ∈ 𝑉)
25 simplr 768 . . . . . . . . 9 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 βŠ† 𝑦) β†’ 𝑦 ∈ SAlg)
2625adantrl 715 . . . . . . . 8 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑦 ∈ SAlg)
27 simpr 486 . . . . . . . . 9 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 βŠ† 𝑦) β†’ 𝑋 βŠ† 𝑦)
2827adantrl 715 . . . . . . . 8 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑋 βŠ† 𝑦)
29 simprl 770 . . . . . . . 8 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ βˆͺ 𝑦 = βˆͺ 𝑋)
3024, 12, 26, 28, 29salgenss 45052 . . . . . . 7 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ (SalGenβ€˜π‘‹) βŠ† 𝑦)
3122, 30eqsstrd 4021 . . . . . 6 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑆 βŠ† 𝑦)
3231ex 414 . . . . 5 (((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) β†’ ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))
3332ralrimiva 3147 . . . 4 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))
3420, 33jca 513 . . 3 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦)))
3534ex 414 . 2 (πœ‘ β†’ ((SalGenβ€˜π‘‹) = 𝑆 β†’ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))))
364adantr 482 . . . 4 ((πœ‘ ∧ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))) β†’ 𝑋 ∈ 𝑉)
37 simprl1 1219 . . . 4 ((πœ‘ ∧ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))) β†’ 𝑆 ∈ SAlg)
38 simprl2 1220 . . . 4 ((πœ‘ ∧ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))) β†’ βˆͺ 𝑆 = βˆͺ 𝑋)
39 simprl3 1221 . . . 4 ((πœ‘ ∧ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))) β†’ 𝑋 βŠ† 𝑆)
40 unieq 4920 . . . . . . . . . . . . . . 15 (𝑦 = 𝑀 β†’ βˆͺ 𝑦 = βˆͺ 𝑀)
4140eqeq1d 2735 . . . . . . . . . . . . . 14 (𝑦 = 𝑀 β†’ (βˆͺ 𝑦 = βˆͺ 𝑋 ↔ βˆͺ 𝑀 = βˆͺ 𝑋))
42 sseq2 4009 . . . . . . . . . . . . . 14 (𝑦 = 𝑀 β†’ (𝑋 βŠ† 𝑦 ↔ 𝑋 βŠ† 𝑀))
4341, 42anbi12d 632 . . . . . . . . . . . . 13 (𝑦 = 𝑀 β†’ ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) ↔ (βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀)))
44 sseq2 4009 . . . . . . . . . . . . 13 (𝑦 = 𝑀 β†’ (𝑆 βŠ† 𝑦 ↔ 𝑆 βŠ† 𝑀))
4543, 44imbi12d 345 . . . . . . . . . . . 12 (𝑦 = 𝑀 β†’ (((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦) ↔ ((βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀) β†’ 𝑆 βŠ† 𝑀)))
4645cbvralvw 3235 . . . . . . . . . . 11 (βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦) ↔ βˆ€π‘€ ∈ SAlg ((βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀) β†’ 𝑆 βŠ† 𝑀))
4746biimpi 215 . . . . . . . . . 10 (βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦) β†’ βˆ€π‘€ ∈ SAlg ((βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀) β†’ 𝑆 βŠ† 𝑀))
4847adantr 482 . . . . . . . . 9 ((βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦) ∧ 𝑀 ∈ SAlg) β†’ βˆ€π‘€ ∈ SAlg ((βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀) β†’ 𝑆 βŠ† 𝑀))
49 simpr 486 . . . . . . . . 9 ((βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦) ∧ 𝑀 ∈ SAlg) β†’ 𝑀 ∈ SAlg)
5048, 49jca 513 . . . . . . . 8 ((βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦) ∧ 𝑀 ∈ SAlg) β†’ (βˆ€π‘€ ∈ SAlg ((βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀) β†’ 𝑆 βŠ† 𝑀) ∧ 𝑀 ∈ SAlg))
51503ad2antr1 1189 . . . . . . 7 ((βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦) ∧ (𝑀 ∈ SAlg ∧ βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀)) β†’ (βˆ€π‘€ ∈ SAlg ((βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀) β†’ 𝑆 βŠ† 𝑀) ∧ 𝑀 ∈ SAlg))
52 3simpc 1151 . . . . . . . 8 ((𝑀 ∈ SAlg ∧ βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀) β†’ (βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀))
5352adantl 483 . . . . . . 7 ((βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦) ∧ (𝑀 ∈ SAlg ∧ βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀)) β†’ (βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀))
54 rspa 3246 . . . . . . 7 ((βˆ€π‘€ ∈ SAlg ((βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀) β†’ 𝑆 βŠ† 𝑀) ∧ 𝑀 ∈ SAlg) β†’ ((βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀) β†’ 𝑆 βŠ† 𝑀))
5551, 53, 54sylc 65 . . . . . 6 ((βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦) ∧ (𝑀 ∈ SAlg ∧ βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀)) β†’ 𝑆 βŠ† 𝑀)
5655adantll 713 . . . . 5 ((((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦)) ∧ (𝑀 ∈ SAlg ∧ βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀)) β†’ 𝑆 βŠ† 𝑀)
5756adantll 713 . . . 4 (((πœ‘ ∧ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))) ∧ (𝑀 ∈ SAlg ∧ βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀)) β†’ 𝑆 βŠ† 𝑀)
5836, 37, 38, 39, 57issalgend 45054 . . 3 ((πœ‘ ∧ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))) β†’ (SalGenβ€˜π‘‹) = 𝑆)
5958ex 414 . 2 (πœ‘ β†’ (((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦)) β†’ (SalGenβ€˜π‘‹) = 𝑆))
6035, 59impbid 211 1 (πœ‘ β†’ ((SalGenβ€˜π‘‹) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆) ∧ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949  βˆͺ cuni 4909  β€˜cfv 6544  SAlgcsalg 45024  SalGencsalgen 45028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-salg 45025  df-salgen 45029
This theorem is referenced by:  unisalgen2  45070
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