Theorem unisalgen2 46352

Description: The union of a set belongs is equal to the union of the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
unisalgen2.x	⊢ (𝜑 → 𝐴 ∈ 𝑉)
unisalgen2.s	⊢ 𝑆 = (SalGen‘𝐴)

Assertion

Ref	Expression
unisalgen2	⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐴)

Proof of Theorem unisalgen2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unisalgen2.s	. . . . . 6 ⊢ 𝑆 = (SalGen‘𝐴)
2	1	eqcomi 2738	. . . . 5 ⊢ (SalGen‘𝐴) = 𝑆
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (SalGen‘𝐴) = 𝑆)
4		unisalgen2.x	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5	4	dfsalgen2 46339	. . . 4 ⊢ (𝜑 → ((SalGen‘𝐴) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥))))
6	3, 5	mpbid 232	. . 3 ⊢ (𝜑 → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥)))
7	6	simpld 494	. 2 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆))
8	7	simp2d 1143	1 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3914  ∪ cuni 4871  ‘cfv 6511  SAlgcsalg 46306  SalGencsalgen 46310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-salg 46307  df-salgen 46311

This theorem is referenced by:  incsmf  46740  decsmf  46765