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Theorem salgenuni 45958
Description: The base set of the sigma-algebra generated by a set is the union of the set itself. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
salgenuni.x (𝜑𝑋𝑉)
salgenuni.s 𝑆 = (SalGen‘𝑋)
salgenuni.u 𝑈 = 𝑋
Assertion
Ref Expression
salgenuni (𝜑 𝑆 = 𝑈)

Proof of Theorem salgenuni
Dummy variables 𝑡 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 salgenuni.s . . . . 5 𝑆 = (SalGen‘𝑋)
21a1i 11 . . . 4 (𝜑𝑆 = (SalGen‘𝑋))
3 salgenuni.x . . . . 5 (𝜑𝑋𝑉)
4 salgenval 45942 . . . . 5 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
53, 4syl 17 . . . 4 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
62, 5eqtrd 2766 . . 3 (𝜑𝑆 = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
76unieqd 4926 . 2 (𝜑 𝑆 = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
8 ssrab2 4076 . . . 4 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg
98a1i 11 . . 3 (𝜑 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg)
10 salgenn0 45952 . . . 4 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
113, 10syl 17 . . 3 (𝜑 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
12 unieq 4924 . . . . . . . . . 10 (𝑠 = 𝑡 𝑠 = 𝑡)
1312eqeq1d 2728 . . . . . . . . 9 (𝑠 = 𝑡 → ( 𝑠 = 𝑋 𝑡 = 𝑋))
14 sseq2 4006 . . . . . . . . 9 (𝑠 = 𝑡 → (𝑋𝑠𝑋𝑡))
1513, 14anbi12d 630 . . . . . . . 8 (𝑠 = 𝑡 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑡 = 𝑋𝑋𝑡)))
1615elrab 3681 . . . . . . 7 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1716biimpi 215 . . . . . 6 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1817simprld 770 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑋)
19 salgenuni.u . . . . . . 7 𝑈 = 𝑋
2019eqcomi 2735 . . . . . 6 𝑋 = 𝑈
2120a1i 11 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋 = 𝑈)
2218, 21eqtrd 2766 . . . 4 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑈)
2322adantl 480 . . 3 ((𝜑𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑡 = 𝑈)
249, 11, 23intsaluni 45950 . 2 (𝜑 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} = 𝑈)
257, 24eqtrd 2766 1 (𝜑 𝑆 = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wne 2930  {crab 3419  wss 3947  c0 4325   cuni 4913   cint 4954  cfv 6554  SAlgcsalg 45929  SalGencsalgen 45933
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 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562  df-salg 45930  df-salgen 45934
This theorem is referenced by:  unisalgen  45961  dfsalgen2  45962
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