Theorem unisalgen2 42657

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 2830	. . . . 5 ⊢ (SalGen‘𝐴) = 𝑆
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (SalGen‘𝐴) = 𝑆)
4		unisalgen2.x	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5	4	dfsalgen2 42644	. . . 4 ⊢ (𝜑 → ((SalGen‘𝐴) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥))))
6	3, 5	mpbid 234	. . 3 ⊢ (𝜑 → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥)))
7	6	simpld 497	. 2 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆))
8	7	simp2d 1139	1 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ⊆ wss 3936  ∪ cuni 4838  ‘cfv 6355  SAlgcsalg 42613  SalGencsalgen 42617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-int 4877  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-salg 42614  df-salgen 42618

This theorem is referenced by:  incsmf  43039  decsmf  43063