Theorem sssigagen2 33935

Description: A subset of the generating set is also a subset of the generated sigma-algebra. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Assertion

Ref	Expression
sssigagen2	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ (sigaGen‘𝐴))

Proof of Theorem sssigagen2

Step	Hyp	Ref	Expression
1		simpr 483	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
2		sssigagen 33934	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))
3	2	adantr 479	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ (sigaGen‘𝐴))
4	1, 3	sstrd 3989	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ (sigaGen‘𝐴))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098   ⊆ wss 3946  ‘cfv 6553  sigaGencsigagen 33927

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 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745

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 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-iota 6505  df-fun 6555  df-fv 6561  df-siga 33898  df-sigagen 33928

This theorem is referenced by:  sxbrsigalem5  34078