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Theorem salgenuni 41302
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 41288 . . . . 5 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
53, 4syl 17 . . . 4 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
62, 5eqtrd 2837 . . 3 (𝜑𝑆 = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
76unieqd 4642 . 2 (𝜑 𝑆 = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
8 ssrab2 3887 . . . 4 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg
98a1i 11 . . 3 (𝜑 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg)
10 salgenn0 41296 . . . 4 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
113, 10syl 17 . . 3 (𝜑 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
12 unieq 4640 . . . . . . . . . 10 (𝑠 = 𝑡 𝑠 = 𝑡)
1312eqeq1d 2805 . . . . . . . . 9 (𝑠 = 𝑡 → ( 𝑠 = 𝑋 𝑡 = 𝑋))
14 sseq2 3827 . . . . . . . . 9 (𝑠 = 𝑡 → (𝑋𝑠𝑋𝑡))
1513, 14anbi12d 625 . . . . . . . 8 (𝑠 = 𝑡 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑡 = 𝑋𝑋𝑡)))
1615elrab 3560 . . . . . . 7 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1716biimpi 208 . . . . . 6 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1817simprld 789 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑋)
19 salgenuni.u . . . . . . 7 𝑈 = 𝑋
2019eqcomi 2812 . . . . . 6 𝑋 = 𝑈
2120a1i 11 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋 = 𝑈)
2218, 21eqtrd 2837 . . . 4 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑈)
2322adantl 474 . . 3 ((𝜑𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑡 = 𝑈)
249, 11, 23intsaluni 41294 . 2 (𝜑 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} = 𝑈)
257, 24eqtrd 2837 1 (𝜑 𝑆 = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wne 2975  {crab 3097  wss 3773  c0 4119   cuni 4632   cint 4671  cfv 6105  SAlgcsalg 41275  SalGencsalgen 41279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-rab 3102  df-v 3391  df-sbc 3638  df-csb 3733  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-pw 4355  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-int 4672  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-iota 6068  df-fun 6107  df-fv 6113  df-salg 41276  df-salgen 41280
This theorem is referenced by:  unisalgen  41305  dfsalgen2  41306
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