Theorem unisalgen2 45804

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 2734	. . . . 5 ⊢ (SalGen‘𝐴) = 𝑆
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (SalGen‘𝐴) = 𝑆)
4		unisalgen2.x	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5	4	dfsalgen2 45791	. . . 4 ⊢ (𝜑 → ((SalGen‘𝐴) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥))))
6	3, 5	mpbid 231	. . 3 ⊢ (𝜑 → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥)))
7	6	simpld 493	. 2 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆))
8	7	simp2d 1140	1 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3051   ⊆ wss 3940  ∪ cuni 4903  ‘cfv 6542  SAlgcsalg 45758  SalGencsalgen 45762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-salg 45759  df-salgen 45763

This theorem is referenced by:  incsmf  46192  decsmf  46217