Theorem unisalgen2 47185

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 2769	. . . . 5 ⊢ (SalGen‘𝐴) = 𝑆
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (SalGen‘𝐴) = 𝑆)
4		unisalgen2.x	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5	4	dfsalgen2 47172	. . . 4 ⊢ (𝜑 → ((SalGen‘𝐴) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥))))
6	3, 5	mpbid 235	. . 3 ⊢ (𝜑 → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥)))
7	6	simpld 500	. 2 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆))
8	7	simp2d 1161	1 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  ∪ cuni 4867  ‘cfv 6533  SAlgcsalg 47139  SalGencsalgen 47143

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-salg 47140  df-salgen 47144

This theorem is used by:  incsmf  47573  decsmf  47598