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Theorem dfsalgen2 45047
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 2738 . . . . . . 7 ((SalGenβ€˜π‘‹) = 𝑆 β†’ 𝑆 = (SalGenβ€˜π‘‹))
32adantl 482 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ 𝑆 = (SalGenβ€˜π‘‹))
4 dfsalgen2.1 . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ 𝑉)
5 salgencl 45038 . . . . . . . 8 (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
64, 5syl 17 . . . . . . 7 (πœ‘ β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
76adantr 481 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
83, 7eqeltrd 2833 . . . . 5 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ 𝑆 ∈ SAlg)
9 unieq 4919 . . . . . . 7 ((SalGenβ€˜π‘‹) = 𝑆 β†’ βˆͺ (SalGenβ€˜π‘‹) = βˆͺ 𝑆)
109adantl 482 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ βˆͺ (SalGenβ€˜π‘‹) = βˆͺ 𝑆)
114adantr 481 . . . . . . 7 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ 𝑋 ∈ 𝑉)
12 eqid 2732 . . . . . . 7 (SalGenβ€˜π‘‹) = (SalGenβ€˜π‘‹)
13 eqid 2732 . . . . . . 7 βˆͺ 𝑋 = βˆͺ 𝑋
1411, 12, 13salgenuni 45043 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ βˆͺ (SalGenβ€˜π‘‹) = βˆͺ 𝑋)
1510, 14eqtr3d 2774 . . . . 5 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ βˆͺ 𝑆 = βˆͺ 𝑋)
1612sssalgen 45041 . . . . . . 7 (𝑋 ∈ 𝑉 β†’ 𝑋 βŠ† (SalGenβ€˜π‘‹))
1711, 16syl 17 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ 𝑋 βŠ† (SalGenβ€˜π‘‹))
18 simpr 485 . . . . . 6 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ (SalGenβ€˜π‘‹) = 𝑆)
1917, 18sseqtrd 4022 . . . . 5 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ 𝑋 βŠ† 𝑆)
208, 15, 193jca 1128 . . . 4 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ (𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑆))
213ad2antrr 724 . . . . . . . 8 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 = (SalGenβ€˜π‘‹))
2221adantrl 714 . . . . . . 7 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑆 = (SalGenβ€˜π‘‹))
2311ad2antrr 724 . . . . . . . . 9 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 βŠ† 𝑦) β†’ 𝑋 ∈ 𝑉)
2423adantrl 714 . . . . . . . 8 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑋 ∈ 𝑉)
25 simplr 767 . . . . . . . . 9 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 βŠ† 𝑦) β†’ 𝑦 ∈ SAlg)
2625adantrl 714 . . . . . . . 8 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑦 ∈ SAlg)
27 simpr 485 . . . . . . . . 9 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋 βŠ† 𝑦) β†’ 𝑋 βŠ† 𝑦)
2827adantrl 714 . . . . . . . 8 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑋 βŠ† 𝑦)
29 simprl 769 . . . . . . . 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 45042 . . . . . . 7 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ (SalGenβ€˜π‘‹) βŠ† 𝑦)
3122, 30eqsstrd 4020 . . . . . 6 ((((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑆 βŠ† 𝑦)
3231ex 413 . . . . 5 (((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) ∧ 𝑦 ∈ SAlg) β†’ ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))
3332ralrimiva 3146 . . . 4 ((πœ‘ ∧ (SalGenβ€˜π‘‹) = 𝑆) β†’ βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦))
3420, 33jca 512 . . 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 413 . 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 481 . . . 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 1218 . . . 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 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†’ βˆͺ 𝑆 = βˆͺ 𝑋)
39 simprl3 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†’ 𝑋 βŠ† 𝑆)
40 unieq 4919 . . . . . . . . . . . . . . 15 (𝑦 = 𝑀 β†’ βˆͺ 𝑦 = βˆͺ 𝑀)
4140eqeq1d 2734 . . . . . . . . . . . . . 14 (𝑦 = 𝑀 β†’ (βˆͺ 𝑦 = βˆͺ 𝑋 ↔ βˆͺ 𝑀 = βˆͺ 𝑋))
42 sseq2 4008 . . . . . . . . . . . . . 14 (𝑦 = 𝑀 β†’ (𝑋 βŠ† 𝑦 ↔ 𝑋 βŠ† 𝑀))
4341, 42anbi12d 631 . . . . . . . . . . . . 13 (𝑦 = 𝑀 β†’ ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) ↔ (βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀)))
44 sseq2 4008 . . . . . . . . . . . . 13 (𝑦 = 𝑀 β†’ (𝑆 βŠ† 𝑦 ↔ 𝑆 βŠ† 𝑀))
4543, 44imbi12d 344 . . . . . . . . . . . 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 3234 . . . . . . . . . . 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 481 . . . . . . . . 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 485 . . . . . . . . 9 ((βˆ€π‘¦ ∈ SAlg ((βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦) β†’ 𝑆 βŠ† 𝑦) ∧ 𝑀 ∈ SAlg) β†’ 𝑀 ∈ SAlg)
5048, 49jca 512 . . . . . . . 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 1188 . . . . . . 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 1150 . . . . . . . 8 ((𝑀 ∈ SAlg ∧ βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀) β†’ (βˆͺ 𝑀 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑀))
5352adantl 482 . . . . . . 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 3245 . . . . . . 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 712 . . . . 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 712 . . . 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 45044 . . 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 413 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948  βˆͺ cuni 4908  β€˜cfv 6543  SAlgcsalg 45014  SalGencsalgen 45018
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-salg 45015  df-salgen 45019
This theorem is referenced by:  unisalgen2  45060
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