Theorem unisalgen2 46608

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 2745	. . . . 5 ⊢ (SalGen‘𝐴) = 𝑆
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (SalGen‘𝐴) = 𝑆)
4		unisalgen2.x	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5	4	dfsalgen2 46595	. . . 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 1541   ∈ wcel 2113  ∀wral 3051   ⊆ wss 3901  ∪ cuni 4863  ‘cfv 6492  SAlgcsalg 46562  SalGencsalgen 46566

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-salg 46563  df-salgen 46567

This theorem is referenced by:  incsmf  46996  decsmf  47021